Twierdzenie Bolzana-Weierstrassa – Wikipedia, wolna encyklopedia

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="pl" dir="ltr"> <head> <meta charset="UTF-8"> <title>Twierdzenie Bolzana-Weierstrassa – Wikipedia, wolna encyklopedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )plwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":[",\t."," \t,"],"wgDigitTransformTable":["",""], "wgDefaultDateFormat":"dmy","wgMonthNames":["","styczeń","luty","marzec","kwiecień","maj","czerwiec","lipiec","sierpień","wrzesień","październik","listopad","grudzień"],"wgRequestId":"8f71f6e6-9fe5-455a-809e-3e2c54108077","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Twierdzenie_Bolzana-Weierstrassa","wgTitle":"Twierdzenie Bolzana-Weierstrassa","wgCurRevisionId":74231318,"wgRevisionId":74231318,"wgArticleId":495132,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Szablon cytowania książki – brak numeru strony","Twierdzenia o ciągach rzeczywistych"],"wgPageViewLanguage":"pl","wgPageContentLanguage":"pl","wgPageContentModel":"wikitext","wgRelevantPageName":"Twierdzenie_Bolzana-Weierstrassa","wgRelevantArticleId":495132,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia", "wgCiteReferencePreviewsActive":true,"wgFlaggedRevsParams":{"tags":{"accuracy":{"levels":1}}},"wgStableRevisionId":74231318,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"pl","pageLanguageDir":"ltr","pageVariantFallbacks":"pl"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":9000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q468391","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false, "wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.gadget.wikiflex":"ready","ext.gadget.infobox":"ready","ext.gadget.hlist":"ready","ext.gadget.darkmode-overrides":"ready","ext.gadget.small-references":"ready","ext.gadget.citation-access-info":"ready","ext.gadget.sprawdz-problemy-szablony":"ready","ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.flaggedRevs.basic":"ready","mediawiki.codex.messagebox.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=[ "ext.cite.ux-enhancements","ext.scribunto.logs","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.flaggedRevs.advanced","ext.gadget.ll-script-loader","ext.gadget.veKeepParameters","ext.gadget.szablon-galeria","ext.gadget.NavFrame","ext.gadget.citoid-overrides","ext.gadget.maps","ext.gadget.padlock-indicators","ext.gadget.interwiki-langlist","ext.gadget.edit-summaries","ext.gadget.edit-first-section","ext.gadget.wikibugs","ext.gadget.map-toggler","ext.gadget.narrowFootnoteColumns","ext.gadget.WDsearch","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints", "ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=pl&modules=ext.cite.styles%7Cext.flaggedRevs.basic%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cmediawiki.codex.messagebox.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=pl&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=pl&modules=ext.gadget.citation-access-info%2Cdarkmode-overrides%2Chlist%2Cinfobox%2Csmall-references%2Csprawdz-problemy-szablony%2Cwikiflex&only=styles&skin=vector-2022"> <link rel="stylesheet" href="/w/load.php?lang=pl&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Twierdzenie Bolzana-Weierstrassa – Wikipedia, wolna encyklopedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//pl.m.wikipedia.org/wiki/Twierdzenie_Bolzana-Weierstrassa"> <link rel="alternate" type="application/x-wiki" title="Edytuj" href="/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (pl)"> <link rel="EditURI" type="application/rsd+xml" href="//pl.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://pl.wikipedia.org/wiki/Twierdzenie_Bolzana-Weierstrassa"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.pl"> <link rel="alternate" type="application/atom+xml" title="Kanał Atom Wikipedii" href="/w/index.php?title=Specjalna:Ostatnie_zmiany&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Twierdzenie_Bolzana-Weierstrassa rootpage-Twierdzenie_Bolzana-Weierstrassa skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Przejdź do zawartości</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Witryna"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Menu główne" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Menu główne</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Menu główne</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">przypnij</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">ukryj</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Nawigacja </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Wikipedia:Strona_g%C5%82%C3%B3wna" title="Przejdź na stronę główną [z]" accesskey="z"><span>Strona główna</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Specjalna:Losowa_strona" title="Załaduj losową stronę [x]" accesskey="x"><span>Losuj artykuł</span></a></li><li id="n-Kategorie" class="mw-list-item"><a href="/wiki/Portal:Kategorie_G%C5%82%C3%B3wne"><span>Kategorie artykułów</span></a></li><li id="n-Featured-articles" class="mw-list-item"><a href="/wiki/Wikipedia:Wyr%C3%B3%C5%BCniona_zawarto%C5%9B%C4%87_Wikipedii"><span>Najlepsze artykuły</span></a></li><li id="n-FAQ" class="mw-list-item"><a href="/wiki/Pomoc:FAQ"><span>Częste pytania (FAQ)</span></a></li> </ul> </div> </div> <div id="p-zmiany" class="vector-menu mw-portlet mw-portlet-zmiany" > <div class="vector-menu-heading"> Dla czytelników </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-czytelnicy" class="mw-list-item"><a href="/wiki/Wikipedia:O_Wikipedii"><span>O Wikipedii</span></a></li><li id="n-contact" class="mw-list-item"><a href="/wiki/Wikipedia:Kontakt_z_wikipedystami"><span>Kontakt</span></a></li> </ul> </div> </div> <div id="p-edytorzy" class="vector-menu mw-portlet mw-portlet-edytorzy" > <div class="vector-menu-heading"> Dla wikipedystów </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-pierwsze-kroki" class="mw-list-item"><a href="/wiki/Pomoc:Pierwsze_kroki"><span>Pierwsze kroki</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Portal_wikipedyst%C3%B3w" title="O projekcie – co możesz zrobić, gdzie możesz znaleźć informacje"><span>Portal wikipedystów</span></a></li><li id="n-Noticeboard" class="mw-list-item"><a href="/wiki/Wikipedia:Tablica_og%C5%82osze%C5%84"><span>Ogłoszenia</span></a></li><li id="n-Guidelines" class="mw-list-item"><a href="/wiki/Wikipedia:Zasady"><span>Zasady</span></a></li><li id="n-helppage-name" class="mw-list-item"><a href="/wiki/Pomoc:Spis_tre%C5%9Bci"><span>Pomoc</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Specjalna:Ostatnie_zmiany" title="Lista ostatnich zmian w Wikipedii. [r]" accesskey="r"><span>Ostatnie zmiany</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Wikipedia:Strona_g%C5%82%C3%B3wna" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="wolna encyklopedia" src="/static/images/mobile/copyright/wikipedia-tagline-pl.svg" width="120" height="13" style="width: 7.5em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Specjalna:Szukaj" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Przeszukaj Wikipedię [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Szukaj</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Przeszukaj Wikipedię" aria-label="Przeszukaj Wikipedię" autocapitalize="sentences" title="Przeszukaj Wikipedię [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Specjalna:Szukaj"> </div> <button class="cdx-button cdx-search-input__end-button">Szukaj</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Narzędzia osobiste"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Wygląd"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Zmień rozmiar czcionki, szerokość oraz kolorystykę strony" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Wygląd" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Wygląd</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_pl.wikipedia.org&uselang=pl" class=""><span>Wspomóż Wikipedię</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Specjalna:Utw%C3%B3rz_konto&returnto=Twierdzenie+Bolzana-Weierstrassa" title="Zachęcamy do stworzenia konta i zalogowania, ale nie jest to obowiązkowe." class=""><span>Utwórz konto</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Specjalna:Zaloguj&returnto=Twierdzenie+Bolzana-Weierstrassa" title="Zachęcamy do zalogowania się, choć nie jest to obowiązkowe. [o]" accesskey="o" class=""><span>Zaloguj się</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Więcej opcji" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Narzędzia osobiste" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Narzędzia osobiste</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Menu użytkownika" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_pl.wikipedia.org&uselang=pl"><span>Wspomóż Wikipedię</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Specjalna:Utw%C3%B3rz_konto&returnto=Twierdzenie+Bolzana-Weierstrassa" title="Zachęcamy do stworzenia konta i zalogowania, ale nie jest to obowiązkowe."><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Utwórz konto</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Specjalna:Zaloguj&returnto=Twierdzenie+Bolzana-Weierstrassa" title="Zachęcamy do zalogowania się, choć nie jest to obowiązkowe. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Zaloguj się</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Strony dla anonimowych edytorów <a href="/wiki/Pomoc:Pierwsze_kroki" aria-label="Dowiedz się więcej na temat edytowania"><span>dowiedz się więcej</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Specjalna:M%C3%B3j_wk%C5%82ad" title="Lista edycji wykonanych z tego adresu IP [y]" accesskey="y"><span>Edycje</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Specjalna:Moja_dyskusja" title="Dyskusja użytkownika dla tego adresu IP [n]" accesskey="n"><span>Dyskusja</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Witryna"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Spis treści" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Spis treści</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">przypnij</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ukryj</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Początek</div> </a> </li> <li id="toc-Twierdzenie" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Twierdzenie"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Twierdzenie</span> </div> </a> <ul id="toc-Twierdzenie-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dowody" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dowody"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Dowody</span> </div> </a> <button aria-controls="toc-Dowody-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Przełącz podsekcję Dowody</span> </button> <ul id="toc-Dowody-sublist" class="vector-toc-list"> <li id="toc-Pierwszy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pierwszy"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Pierwszy</span> </div> </a> <ul id="toc-Pierwszy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Drugi" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Drugi"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Drugi</span> </div> </a> <ul id="toc-Drugi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Trzeci" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Trzeci"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Trzeci</span> </div> </a> <ul id="toc-Trzeci-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Uwagi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Uwagi"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Uwagi</span> </div> </a> <ul id="toc-Uwagi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Literatura_dodatkowa" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Literatura_dodatkowa"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Literatura dodatkowa</span> </div> </a> <ul id="toc-Literatura_dodatkowa-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Linki_zewnętrzne" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Linki_zewnętrzne"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Linki zewnętrzne</span> </div> </a> <ul id="toc-Linki_zewnętrzne-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Spis treści" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Przełącz stan spisu treści" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Przełącz stan spisu treści</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Twierdzenie Bolzana-Weierstrassa</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Przejdź do artykułu w innym języku. Treść dostępna w 33 językach" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-33" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">33 języki</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%A8%D8%B1%D9%87%D9%86%D8%A9_%D8%A8%D9%88%D9%84%D8%B2%D8%A7%D9%86%D9%88-%D9%81%D8%A7%D9%8A%D8%B1%D8%B4%D8%AA%D8%B1%D8%A7%D8%B3" title="مبرهنة بولزانو-فايرشتراس – arabski" lang="ar" hreflang="ar" data-title="مبرهنة بولزانو-فايرشتراس" data-language-autonym="العربية" data-language-local-name="arabski" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Boltsano-Veyer%C5%9Ftrass_teoremi" title="Boltsano-Veyerştrass teoremi – azerbejdżański" lang="az" hreflang="az" data-title="Boltsano-Veyerştrass teoremi" data-language-autonym="Azərbaycanca" data-language-local-name="azerbejdżański" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BD%D0%B0_%D0%91%D0%BE%D0%BB%D1%86%D0%B0%D0%BD%D0%BE_%E2%80%93_%D0%92%D0%B0%D0%B9%D0%B5%D1%80%D1%89%D1%80%D0%B0%D1%81_(%D0%B7%D0%B0_%D0%B1%D0%B5%D0%B7%D0%BA%D1%80%D0%B0%D0%B9%D0%BD%D0%B8%D1%82%D0%B5_%D1%80%D0%B5%D0%B4%D0%B8%D1%86%D0%B8)" title="Теорема на Болцано – Вайерщрас (за безкрайните редици) – bułgarski" lang="bg" hreflang="bg" data-title="Теорема на Болцано – Вайерщрас (за безкрайните редици)" data-language-autonym="Български" data-language-local-name="bułgarski" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teorema_de_Bolzano-Weierstrass" title="Teorema de Bolzano-Weierstrass – kataloński" lang="ca" hreflang="ca" data-title="Teorema de Bolzano-Weierstrass" data-language-autonym="Català" data-language-local-name="kataloński" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Bolzanova%E2%80%93Weierstrassova_v%C4%9Bta" title="Bolzanova–Weierstrassova věta – czeski" lang="cs" hreflang="cs" data-title="Bolzanova–Weierstrassova věta" data-language-autonym="Čeština" data-language-local-name="czeski" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Theorem_Bolzano-Weierstrass" title="Theorem Bolzano-Weierstrass – walijski" lang="cy" hreflang="cy" data-title="Theorem Bolzano-Weierstrass" data-language-autonym="Cymraeg" data-language-local-name="walijski" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Satz_von_Bolzano-Weierstra%C3%9F" title="Satz von Bolzano-Weierstraß – niemiecki" lang="de" hreflang="de" data-title="Satz von Bolzano-Weierstraß" data-language-autonym="Deutsch" data-language-local-name="niemiecki" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Bolzano-Weierstrassi_teoreem" title="Bolzano-Weierstrassi teoreem – estoński" lang="et" hreflang="et" data-title="Bolzano-Weierstrassi teoreem" data-language-autonym="Eesti" data-language-local-name="estoński" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%98%CE%B5%CF%8E%CF%81%CE%B7%C2%B5%CE%B1_%CE%9C%CF%80%CE%BF%CE%BB%CE%B6%CE%AC%CE%BD%CE%BF-%CE%92%CE%AC%CE%B9%CE%B5%CF%81%CF%83%CF%84%CF%81%CE%B1%CF%82" title="Θεώρηµα Μπολζάνο-Βάιερστρας – grecki" lang="el" hreflang="el" data-title="Θεώρηµα Μπολζάνο-Βάιερστρας" data-language-autonym="Ελληνικά" data-language-local-name="grecki" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Bolzano%E2%80%93Weierstrass_theorem" title="Bolzano–Weierstrass theorem – angielski" lang="en" hreflang="en" data-title="Bolzano–Weierstrass theorem" data-language-autonym="English" data-language-local-name="angielski" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teorema_de_Bolzano-Weierstrass" title="Teorema de Bolzano-Weierstrass – hiszpański" lang="es" hreflang="es" data-title="Teorema de Bolzano-Weierstrass" data-language-autonym="Español" data-language-local-name="hiszpański" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%B6%DB%8C%D9%87_%D8%A8%D9%88%D9%84%D8%B2%D8%A7%D9%86%D9%88-%D9%88%D8%A7%DB%8C%D8%B1%D8%B4%D8%AA%D8%B1%D8%A7%D8%B3" title="قضیه بولزانو-وایرشتراس – perski" lang="fa" hreflang="fa" data-title="قضیه بولزانو-وایرشتراس" data-language-autonym="فارسی" data-language-local-name="perski" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Bolzano-Weierstrass" title="Théorème de Bolzano-Weierstrass – francuski" lang="fr" hreflang="fr" data-title="Théorème de Bolzano-Weierstrass" data-language-autonym="Français" data-language-local-name="francuski" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teorema_de_Bolzano-Weierstrass" title="Teorema de Bolzano-Weierstrass – galicyjski" lang="gl" hreflang="gl" data-title="Teorema de Bolzano-Weierstrass" data-language-autonym="Galego" data-language-local-name="galicyjski" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B3%BC%EC%B0%A8%EB%85%B8-%EB%B0%94%EC%9D%B4%EC%96%B4%EC%8A%88%ED%8A%B8%EB%9D%BC%EC%8A%A4_%EC%A0%95%EB%A6%AC" title="볼차노-바이어슈트라스 정리 – koreański" lang="ko" hreflang="ko" data-title="볼차노-바이어슈트라스 정리" data-language-autonym="한국어" data-language-local-name="koreański" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Bolzano-Weierstrass_setningin" title="Bolzano-Weierstrass setningin – islandzki" lang="is" hreflang="is" data-title="Bolzano-Weierstrass setningin" data-language-autonym="Íslenska" data-language-local-name="islandzki" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teorema_di_Bolzano-Weierstrass" title="Teorema di Bolzano-Weierstrass – włoski" lang="it" hreflang="it" data-title="Teorema di Bolzano-Weierstrass" data-language-autonym="Italiano" data-language-local-name="włoski" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%91%D7%95%D7%9C%D7%A6%D7%90%D7%A0%D7%95-%D7%95%D7%99%D7%99%D7%A8%D7%A9%D7%98%D7%A8%D7%90%D7%A1" title="משפט בולצאנו-ויירשטראס – hebrajski" lang="he" hreflang="he" data-title="משפט בולצאנו-ויירשטראס" data-language-autonym="עברית" data-language-local-name="hebrajski" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Bolzano%E2%80%93Weierstrass-t%C3%A9tel" title="Bolzano–Weierstrass-tétel – węgierski" lang="hu" hreflang="hu" data-title="Bolzano–Weierstrass-tétel" data-language-autonym="Magyar" data-language-local-name="węgierski" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Stelling_van_Bolzano-Weierstrass" title="Stelling van Bolzano-Weierstrass – niderlandzki" lang="nl" hreflang="nl" data-title="Stelling van Bolzano-Weierstrass" data-language-autonym="Nederlands" data-language-local-name="niderlandzki" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%9C%E3%83%AB%E3%83%84%E3%82%A1%E3%83%BC%E3%83%8E%EF%BC%9D%E3%83%AF%E3%82%A4%E3%82%A8%E3%83%AB%E3%82%B7%E3%83%A5%E3%83%88%E3%83%A9%E3%82%B9%E3%81%AE%E5%AE%9A%E7%90%86" title="ボルツァーノ＝ワイエルシュトラスの定理 – japoński" lang="ja" hreflang="ja" data-title="ボルツァーノ＝ワイエルシュトラスの定理" data-language-autonym="日本語" data-language-local-name="japoński" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teorema_de_Bolzano-Weierstrass" title="Teorema de Bolzano-Weierstrass – portugalski" lang="pt" hreflang="pt" data-title="Teorema de Bolzano-Weierstrass" data-language-autonym="Português" data-language-local-name="portugalski" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Teorema_Weierstrass-Bolzano" title="Teorema Weierstrass-Bolzano – rumuński" lang="ro" hreflang="ro" data-title="Teorema Weierstrass-Bolzano" data-language-autonym="Română" data-language-local-name="rumuński" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%91%D0%BE%D0%BB%D1%8C%D1%86%D0%B0%D0%BD%D0%BE_%E2%80%94_%D0%92%D0%B5%D0%B9%D0%B5%D1%80%D1%88%D1%82%D1%80%D0%B0%D1%81%D1%81%D0%B0" title="Теорема Больцано — Вейерштрасса – rosyjski" lang="ru" hreflang="ru" data-title="Теорема Больцано — Вейерштрасса" data-language-autonym="Русский" data-language-local-name="rosyjski" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%91%D0%BE%D0%BB%D1%86%D0%B0%D0%BD%D0%BE-%D0%92%D0%B0%D1%98%D0%B5%D1%80%D1%88%D1%82%D1%80%D0%B0%D1%81%D0%BE%D0%B2%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Болцано-Вајерштрасова теорема – serbski" lang="sr" hreflang="sr" data-title="Болцано-Вајерштрасова теорема" data-language-autonym="Српски / srpski" data-language-local-name="serbski" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Bolzanon%E2%80%93Weierstrassin_lause" title="Bolzanon–Weierstrassin lause – fiński" lang="fi" hreflang="fi" data-title="Bolzanon–Weierstrassin lause" data-language-autonym="Suomi" data-language-local-name="fiński" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Bolzano%E2%80%93Weierstrass_sats" title="Bolzano–Weierstrass sats – szwedzki" lang="sv" hreflang="sv" data-title="Bolzano–Weierstrass sats" data-language-autonym="Svenska" data-language-local-name="szwedzki" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A4%E0%B8%A9%E0%B8%8E%E0%B8%B5%E0%B8%9A%E0%B8%97%E0%B8%9A%E0%B9%87%E0%B8%AD%E0%B8%A5%E0%B8%97%E0%B9%8C%E0%B8%8B%E0%B8%B2%E0%B9%82%E0%B8%99-%E0%B9%84%E0%B8%A7%E0%B9%80%E0%B8%AD%E0%B8%AD%E0%B8%A3%E0%B9%8C%E0%B8%8A%E0%B8%95%E0%B8%A3%E0%B8%B2%E0%B8%AA" title="ทฤษฎีบทบ็อลท์ซาโน-ไวเออร์ชตราส – tajski" lang="th" hreflang="th" data-title="ทฤษฎีบทบ็อลท์ซาโน-ไวเออร์ชตราส" data-language-autonym="ไทย" data-language-local-name="tajski" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Bolzano-Weierstrass_teoremi" title="Bolzano-Weierstrass teoremi – turecki" lang="tr" hreflang="tr" data-title="Bolzano-Weierstrass teoremi" data-language-autonym="Türkçe" data-language-local-name="turecki" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%91%D0%BE%D0%BB%D1%8C%D1%86%D0%B0%D0%BD%D0%BE_%E2%80%94_%D0%92%D0%B5%D1%94%D1%80%D1%88%D1%82%D1%80%D0%B0%D1%81%D1%81%D0%B0" title="Теорема Больцано — Веєрштрасса – ukraiński" lang="uk" hreflang="uk" data-title="Теорема Больцано — Веєрштрасса" data-language-autonym="Українська" data-language-local-name="ukraiński" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%8Bnh_l%C3%BD_Bolzano%E2%80%93Weierstrass" title="Định lý Bolzano–Weierstrass – wietnamski" lang="vi" hreflang="vi" data-title="Định lý Bolzano–Weierstrass" data-language-autonym="Tiếng Việt" data-language-local-name="wietnamski" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%BF%9D%E8%A5%BF%E5%A5%B4-%E8%8F%AF%E5%AF%A6%E6%96%AF%E5%AE%9A%E7%90%86" title="保西奴-華實斯定理 – kantoński" lang="yue" hreflang="yue" data-title="保西奴-華實斯定理" data-language-autonym="粵語" data-language-local-name="kantoński" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%B3%A2%E7%88%BE%E6%9F%A5%E8%AB%BE-%E9%AD%8F%E7%88%BE%E6%96%AF%E7%89%B9%E6%8B%89%E6%96%AF%E5%AE%9A%E7%90%86" title="波爾查諾-魏爾斯特拉斯定理 – chiński" lang="zh" hreflang="zh" data-title="波爾查諾-魏爾斯特拉斯定理" data-language-autonym="中文" data-language-local-name="chiński" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q468391#sitelinks-wikipedia" title="Edytuj linki pomiędzy wersjami językowymi" class="wbc-editpage">Edytuj linki</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Przestrzenie nazw"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Twierdzenie_Bolzana-Weierstrassa" title="Zobacz stronę treści [c]" accesskey="c"><span>Artykuł</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Dyskusja:Twierdzenie_Bolzana-Weierstrassa" rel="discussion" title="Dyskusja o zawartości tej strony [t]" accesskey="t"><span>Dyskusja</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Zmień wariant języka" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">polski</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Widok"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Twierdzenie_Bolzana-Weierstrassa"><span>Czytaj</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&veaction=edit" title="Edytuj tę stronę [v]" accesskey="v"><span>Edytuj</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&action=edit" title="Edycja kodu źródłowego strony [e]" accesskey="e"><span>Edytuj kod źródłowy</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&action=history" title="Starsze wersje tej strony [h]" accesskey="h"><span>Wyświetl historię</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Narzędzia dla stron"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Narzędzia" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Narzędzia</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Narzędzia</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">przypnij</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">ukryj</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Więcej opcji" > <div class="vector-menu-heading"> Działania </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Twierdzenie_Bolzana-Weierstrassa"><span>Czytaj</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&veaction=edit" title="Edytuj tę stronę [v]" accesskey="v"><span>Edytuj</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&action=edit" title="Edycja kodu źródłowego strony [e]" accesskey="e"><span>Edytuj kod źródłowy</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&action=history"><span>Wyświetl historię</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Ogólne </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Specjalna:Linkuj%C4%85ce/Twierdzenie_Bolzana-Weierstrassa" title="Pokaż listę wszystkich stron linkujących do tej strony [j]" accesskey="j"><span>Linkujące</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Specjalna:Zmiany_w_linkowanych/Twierdzenie_Bolzana-Weierstrassa" rel="nofollow" title="Ostatnie zmiany w stronach, do których ta strona linkuje [k]" accesskey="k"><span>Zmiany w linkowanych</span></a></li><li id="t-upload" class="mw-list-item"><a href="//pl.wikipedia.org/wiki/Wikipedia:Prześlij_plik" title="Prześlij pliki [u]" accesskey="u"><span>Prześlij plik</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Specjalna:Strony_specjalne" title="Lista wszystkich stron specjalnych [q]" accesskey="q"><span>Strony specjalne</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&oldid=74231318" title="Stały link do tej wersji tej strony"><span>Link do tej wersji</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&action=info" title="Więcej informacji na temat tej strony"><span>Informacje o tej stronie</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Specjalna:Cytuj&page=Twierdzenie_Bolzana-Weierstrassa&id=74231318&wpFormIdentifier=titleform" title="Informacja o tym jak należy cytować tę stronę"><span>Cytowanie tego artykułu</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Specjalna:Skr%C3%B3%C4%87_adres_URL&url=https%3A%2F%2Fpl.wikipedia.org%2Fwiki%2FTwierdzenie_Bolzana-Weierstrassa"><span>Zobacz skrócony adres URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Specjalna:Kod_QR&url=https%3A%2F%2Fpl.wikipedia.org%2Fwiki%2FTwierdzenie_Bolzana-Weierstrassa"><span>Pobierz kod QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Drukuj lub eksportuj </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Specjalna:Ksi%C4%85%C5%BCka&bookcmd=book_creator&referer=Twierdzenie+Bolzana-Weierstrassa"><span>Utwórz książkę</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Specjalna:DownloadAsPdf&page=Twierdzenie_Bolzana-Weierstrassa&action=show-download-screen"><span>Pobierz jako PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&printable=yes" title="Wersja do wydruku [p]" accesskey="p"><span>Wersja do druku</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> W innych projektach </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Bolzano%E2%80%93Weierstrass_theorem" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q468391" title="Link do powiązanego elementu w repozytorium danych [g]" accesskey="g"><span>Element Wikidanych</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Narzędzia dla stron"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Wygląd"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Wygląd</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">przypnij</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ukryj</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Z Wikipedii, wolnej encyklopedii</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="pl" dir="ltr"><p><b>Twierdzenie Bolzana<sup id="cite_ref-1" class="reference"><a href="#cite_note-1">[a]</a></sup>-Weierstrassa</b> – jeden z podstawowych wyników w <a href="/wiki/Analiza_matematyczna" title="Analiza matematyczna">analizie matematycznej</a>. Mówi ono, że każdy <a href="/wiki/Zbi%C3%B3r_ograniczony" title="Zbiór ograniczony">ograniczony</a> <a href="/wiki/Ci%C4%85g_(matematyka)" title="Ciąg (matematyka)">ciąg</a> liczb <a href="/wiki/Liczby_rzeczywiste" title="Liczby rzeczywiste">rzeczywistych</a> zawiera <a href="/wiki/Podci%C4%85g_(matematyka)" title="Podciąg (matematyka)">podciąg</a> <a href="/wiki/Granica_ci%C4%85gu" title="Granica ciągu">zbieżny</a>. We współczesnym ujęciu oznacza to, że <a href="/wiki/Zbi%C3%B3r_domkni%C4%99ty" title="Zbiór domknięty">domknięte</a> i ograniczone podzbiory prostej rzeczywistej są <a href="/wiki/Przestrze%C5%84_ci%C4%85gowo_zwarta" title="Przestrzeń ciągowo zwarta">ciągowo zwarte</a>. Twierdzenie to jest bezpośrednim wnioskiem z <a href="/wiki/Twierdzenie_Heinego-Borela" title="Twierdzenie Heinego-Borela">twierdzenia Heinego-Borela</a>, głoszącego, że podzbiór prostej jest <a href="/wiki/Przestrze%C5%84_zwarta" title="Przestrzeń zwarta">zwarty</a> wtedy i tylko wtedy, gdy jest on domknięty i ograniczony oraz z równoważności zwartości ze <a href="/wiki/Przestrze%C5%84_ci%C4%85gowo_zwarta" title="Przestrzeń ciągowo zwarta">zwartością ciągową</a> w przestrzeniach <a href="/wiki/Przestrze%C5%84_metryzowalna" title="Przestrzeń metryzowalna">metryzowalnych</a>. </p><p>Twierdzenie było najpierw udowodnione przez czeskiego matematyka <a href="/wiki/Bernard_Bolzano" title="Bernard Bolzano">Bernarda Bolzana</a>, ale jego praca pozostała niezauważona. Twierdzenie było później ponownie odkryte i udowodnione przez niemieckiego matematyka <a href="/wiki/Karl_Weierstra%C3%9F" title="Karl Weierstraß">Karla Weierstrassa</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Twierdzenie">Twierdzenie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&veaction=edit&section=1" title="Edytuj sekcję: Twierdzenie" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&action=edit&section=1" title="Edytuj kod źródłowy sekcji: Twierdzenie"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Z każdego <a href="/wiki/Funkcja_ograniczona" title="Funkcja ograniczona">ograniczonego ciągu</a> liczb rzeczywistych <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c_{n})_{n=0}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c_{n})_{n=0}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/888f007ef3ccc230e04a7c640699f1f52c30df7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.354ex; height:3.009ex;" alt="{\displaystyle (c_{n})_{n=0}^{\infty }}"></span> można wybrać podciąg zbieżny, tzn. można wybrać <a href="/wiki/Funkcja_monotoniczna" title="Funkcja monotoniczna">rosnący ciąg</a> indeksów <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{0},n_{1},n_{2},n_{3},\dots ,n_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{0},n_{1},n_{2},n_{3},\dots ,n_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e72c8129b6538ff1d42d46c10736f326450c3bcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.559ex; height:2.009ex;" alt="{\displaystyle n_{0},n_{1},n_{2},n_{3},\dots ,n_{k}}"></span> tak, że ciąg <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c_{n_{k}})_{k=0}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c_{n_{k}})_{k=0}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f196e0dc48620265894d7489cd87390997e04104" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.083ex; height:3.009ex;" alt="{\displaystyle (c_{n_{k}})_{k=0}^{\infty }}"></span> jest zbieżny. </p> <div class="mw-heading mw-heading2"><h2 id="Dowody">Dowody</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&veaction=edit&section=2" title="Edytuj sekcję: Dowody" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&action=edit&section=2" title="Edytuj kod źródłowy sekcji: Dowody"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Pierwszy">Pierwszy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&veaction=edit&section=3" title="Edytuj sekcję: Pierwszy" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&action=edit&section=3" title="Edytuj kod źródłowy sekcji: Pierwszy"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Załóżmy, że <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c_{n})_{n=0}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c_{n})_{n=0}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/888f007ef3ccc230e04a7c640699f1f52c30df7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.354ex; height:3.009ex;" alt="{\displaystyle (c_{n})_{n=0}^{\infty }}"></span> jest ciągiem liczb rzeczywistych, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}"></span> oraz <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<c_{n}<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<c_{n}<b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6268e54104be720f888acd7d46aea76fefad6adc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.65ex; height:2.509ex;" alt="{\displaystyle a<c_{n}<b}"></span> dla wszystkich <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e59df02a9f67a5da3c220f1244c99a46cc4eb1c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:1.676ex;" alt="{\displaystyle n.}"></span> Indukcyjnie wybieramy liczby <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{k},b_{k}\in [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{k},b_{k}\in [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6564cae84a1c0c8a1a7f4736274f41ce36ee1755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.834ex; height:2.843ex;" alt="{\displaystyle a_{k},b_{k}\in [a,b]}"></span> oraz liczby naturalne <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{k},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c59d35025c9b14ab0c96a74b7a573738996e5de8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.13ex; height:2.009ex;" alt="{\displaystyle n_{k},}"></span> tak że dla każdego <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> mamy </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{0}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{0}=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02baf5a761e6150ab9259218f5e02d58dcda28dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.357ex; height:2.509ex;" alt="{\displaystyle n_{0}=0,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}=a,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}=a,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe86a72049aa4988501e603db8f6e1206ef85f51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.259ex; height:2.009ex;" alt="{\displaystyle a_{0}=a,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0}=b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{0}=b,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/801cf19c3cb285b72a6aa20cbe178397d4199f52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.795ex; height:2.509ex;" alt="{\displaystyle b_{0}=b,}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{k}<n_{k+1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo><</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{k}<n_{k+1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e346c25c056ae60d2708da34ee0e094f7a52bb8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.813ex; height:2.176ex;" alt="{\displaystyle n_{k}<n_{k+1},}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{k}\leqslant a_{k+1}\leqslant c_{n_{k+1}}\leqslant b_{k+1}\leqslant b_{k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⩽</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>⩽</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>⩽</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>⩽</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{k}\leqslant a_{k+1}\leqslant c_{n_{k+1}}\leqslant b_{k+1}\leqslant b_{k},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d00de0e37ddc7694f0b24f060c0651df84497a2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.842ex; height:2.843ex;" alt="{\displaystyle a_{k}\leqslant a_{k+1}\leqslant c_{n_{k+1}}\leqslant b_{k+1}\leqslant b_{k},}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{k}-a_{k}=(b-a)\cdot 2^{-k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{k}-a_{k}=(b-a)\cdot 2^{-k},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae24c337e93111112c5d799ba220badd753b7d65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.076ex; height:3.176ex;" alt="{\displaystyle b_{k}-a_{k}=(b-a)\cdot 2^{-k},}"></span></li> <li>zbiór <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{n:c_{n}\in [a_{k},b_{k}]\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>n</mi> <mo>:</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{n:c_{n}\in [a_{k},b_{k}]\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9db2efc63d1c96c8d69d9878c4e755a4435dc120" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.455ex; height:2.843ex;" alt="{\displaystyle \{n:c_{n}\in [a_{k},b_{k}]\}}"></span> jest nieskończony.</li></ul> <p>Pierwszy warunek powyżej definiuje <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{0},a_{0},b_{0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{0},a_{0},b_{0}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/533ef105b2c7aca2b9ef9f21895f86f55b9882f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.5ex; height:2.509ex;" alt="{\displaystyle n_{0},a_{0},b_{0}.}"></span> Przypuśćmy, że wybraliśmy już <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{k},a_{k},b_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{k},a_{k},b_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0daef33fc00123b08f3dd04f5a9611e2a9fe6e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.956ex; height:2.509ex;" alt="{\displaystyle n_{k},a_{k},b_{k}}"></span> tak, że wymagania sformułowane powyżej są spełnione. Niech <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d={\frac {a_{k}+b_{k}}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d={\frac {a_{k}+b_{k}}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cf1728df1a3e7e7b7c56b7cbd36a660c7f34559" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.043ex; height:5.343ex;" alt="{\displaystyle d={\frac {a_{k}+b_{k}}{2}}.}"></span> Jeśli zbiór <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{n:c_{n}\in [a_{k},d]\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>n</mi> <mo>:</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mi>d</mi> <mo stretchy="false">]</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{n:c_{n}\in [a_{k},d]\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4c1d5a3fad32a508ca56eaa2773f5512a7b0e1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.585ex; height:2.843ex;" alt="{\displaystyle \{n:c_{n}\in [a_{k},d]\}}"></span> jest nieskończony, to połóżmy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{k+1}=a_{k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{k+1}=a_{k},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c7eb3a6cc8f0bb380fb0613672fac09483f538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.483ex; height:2.009ex;" alt="{\displaystyle a_{k+1}=a_{k},}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{k+1}=d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{k+1}=d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85918559b2cd2bfff2630f4d993e1afd0791f636" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.501ex; height:2.509ex;" alt="{\displaystyle b_{k+1}=d}"></span> i wybierzmy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{k+1}>n_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>></mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{k+1}>n_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da91e42db2c18bf41f0491980d10337f91c1a0f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.166ex; height:2.176ex;" alt="{\displaystyle n_{k+1}>n_{k}}"></span> tak że <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{k+1}\leqslant c_{n_{k+1}}\leqslant b_{k+1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>⩽</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>⩽</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{k+1}\leqslant c_{n_{k+1}}\leqslant b_{k+1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d618d35272907cd23c87194a3433a400f2f7f6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.24ex; height:2.843ex;" alt="{\displaystyle a_{k+1}\leqslant c_{n_{k+1}}\leqslant b_{k+1}.}"></span> Jeśli zbiór <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{n:c_{n}\in [a_{k},d]\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>n</mi> <mo>:</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mi>d</mi> <mo stretchy="false">]</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{n:c_{n}\in [a_{k},d]\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4c1d5a3fad32a508ca56eaa2773f5512a7b0e1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.585ex; height:2.843ex;" alt="{\displaystyle \{n:c_{n}\in [a_{k},d]\}}"></span> jest skończony, to wtedy zbiór <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{n:c_{n}\in [d,b_{k}]\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>n</mi> <mo>:</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>d</mi> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{n:c_{n}\in [d,b_{k}]\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8830b3cf6c705a070a42a7178bb8449520faced5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.353ex; height:2.843ex;" alt="{\displaystyle \{n:c_{n}\in [d,b_{k}]\}}"></span> musi być nieskończony. W tym wypadku deklarujemy, że <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{k+1}=d,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>d</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{k+1}=d,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca553a45727d463c783cd430272746498f8aa8a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.38ex; height:2.509ex;" alt="{\displaystyle a_{k+1}=d,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{k+1}=b_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{k+1}=b_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bdfe4d53eaa84cc6087008acdf3ce99656787d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.372ex; height:2.509ex;" alt="{\displaystyle b_{k+1}=b_{k}}"></span> i wybieramy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{k+1}>n_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>></mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{k+1}>n_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da91e42db2c18bf41f0491980d10337f91c1a0f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.166ex; height:2.176ex;" alt="{\displaystyle n_{k+1}>n_{k}}"></span> tak że <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{k+1}\leqslant c_{n_{k+1}}\leqslant b_{k+1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>⩽</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>⩽</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{k+1}\leqslant c_{n_{k+1}}\leqslant b_{k+1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d618d35272907cd23c87194a3433a400f2f7f6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.24ex; height:2.843ex;" alt="{\displaystyle a_{k+1}\leqslant c_{n_{k+1}}\leqslant b_{k+1}.}"></span> </p><p>Po przeprowadzeniu powyższej konstrukcji zauważamy, że ciąg <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c_{n_{k}})_{k=0}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c_{n_{k}})_{k=0}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f196e0dc48620265894d7489cd87390997e04104" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.083ex; height:3.009ex;" alt="{\displaystyle (c_{n_{k}})_{k=0}^{\infty }}"></span> jest <a href="/wiki/Ci%C4%85g_Cauchy%E2%80%99ego" title="Ciąg Cauchy’ego">ciągiem Cauchy’ego</a>, a więc wobec <a href="/wiki/Przestrze%C5%84_zupe%C5%82na" title="Przestrzeń zupełna">zupełności</a> prostej rzeczywistej jest on zbieżny. </p> <div class="mw-heading mw-heading3"><h3 id="Drugi">Drugi</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&veaction=edit&section=4" title="Edytuj sekcję: Drugi" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&action=edit&section=4" title="Edytuj kod źródłowy sekcji: Drugi"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Załóżmy, że <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c_{n})_{n=0}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c_{n})_{n=0}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/888f007ef3ccc230e04a7c640699f1f52c30df7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.354ex; height:3.009ex;" alt="{\displaystyle (c_{n})_{n=0}^{\infty }}"></span> jest ograniczonym ciągiem liczb rzeczywistych i niech <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=\inf\{c_{n}:n\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">inf</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>:</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=\inf\{c_{n}:n\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da18234a8dc24eae4aeb46f54389b593ed85abfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.861ex; height:2.843ex;" alt="{\displaystyle L=\inf\{c_{n}:n\},}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=\sup\{c_{n}:n\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">sup</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>:</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=\sup\{c_{n}:n\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92908d2b9dc1b2e2435f5f9f154653a38be7c0f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.893ex; height:2.843ex;" alt="{\displaystyle R=\sup\{c_{n}:n\},}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=(R+L)/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo>+</mo> <mi>L</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=(R+L)/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72dd80100c6cf7c0fa0127093fba1260e21952e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.862ex; height:2.843ex;" alt="{\displaystyle M=(R+L)/2}"></span> i niech <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =[L,R].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mi>L</mi> <mo>,</mo> <mi>R</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =[L,R].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/923e847fa7bd10b0204af86da06080abb445d479" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.356ex; height:2.843ex;" alt="{\displaystyle \Delta =[L,R].}"></span> </p><p>Niech teraz <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{\epsilon }=[L_{\epsilon },R_{\epsilon }]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{\epsilon }=[L_{\epsilon },R_{\epsilon }]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b562b9568404e30fd58a0263ed5265b422c9010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.408ex; height:2.843ex;" alt="{\displaystyle \Delta _{\epsilon }=[L_{\epsilon },R_{\epsilon }]}"></span> będzie rodziną podprzedziałów przedziału <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }"></span> indeksowaną skończonymi ciągami zero-jedynkowymi określoną wzorami: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{\langle 0\rangle }=[L,M],\Delta _{\langle 1\rangle }=[M,R]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <mi>L</mi> <mo>,</mo> <mi>M</mi> <mo stretchy="false">]</mo> <mo>,</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨</mo> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <mi>M</mi> <mo>,</mo> <mi>R</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{\langle 0\rangle }=[L,M],\Delta _{\langle 1\rangle }=[M,R]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10c610a25e21d423e3501534a61476794d36b92f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:28.656ex; height:3.176ex;" alt="{\displaystyle \Delta _{\langle 0\rangle }=[L,M],\Delta _{\langle 1\rangle }=[M,R]}"></span> oraz <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{\epsilon \langle 0\rangle }=[L_{\epsilon },M_{\epsilon }]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{\epsilon \langle 0\rangle }=[L_{\epsilon },M_{\epsilon }]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f1f5a106c4d405b72a9fedc99d0a3b1c9192b14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16ex; height:3.176ex;" alt="{\displaystyle \Delta _{\epsilon \langle 0\rangle }=[L_{\epsilon },M_{\epsilon }]}"></span> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{\epsilon \langle 1\rangle }=[M_{\epsilon },R_{\epsilon }],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> <mo fence="false" stretchy="false">⟨</mo> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{\epsilon \langle 1\rangle }=[M_{\epsilon },R_{\epsilon }],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8b9b98a4d4b5aa8e537bcda93d0aca359298db3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.828ex; height:3.176ex;" alt="{\displaystyle \Delta _{\epsilon \langle 1\rangle }=[M_{\epsilon },R_{\epsilon }],}"></span></dd></dl> <p>gdzie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\epsilon }=(L_{\epsilon }+R_{\epsilon })/2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\epsilon }=(L_{\epsilon }+R_{\epsilon })/2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a85b0cc9a2ff6226100219d541707fb8b1b5001" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.02ex; height:2.843ex;" alt="{\displaystyle M_{\epsilon }=(L_{\epsilon }+R_{\epsilon })/2.}"></span> </p><p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/Plik:Descending_intervals.svg" class="mw-file-description" title="Konstrukcja rodziny przedziałów '"`UNIQ--postMath-0000002C-QINU`"'"><img alt="Konstrukcja rodziny przedziałów '"`UNIQ--postMath-0000002C-QINU`"'" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Descending_intervals.svg/631px-Descending_intervals.svg.png" decoding="async" width="631" height="372" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Descending_intervals.svg/947px-Descending_intervals.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/40/Descending_intervals.svg/1262px-Descending_intervals.svg.png 2x" data-file-width="631" data-file-height="372" /></a></span> </p><p>Łatwo zauważyć, że długość przedziału <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{\epsilon }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{\epsilon }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e65c0f0663b91cf3fc4bdf78f66c3267ea95eb80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.836ex; height:2.509ex;" alt="{\displaystyle \Delta _{\epsilon }}"></span> równa jest <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (R-L)/2^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>R</mi> <mo>−</mo> <mi>L</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (R-L)/2^{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1df41780e7604d5b95d3fac79c3aeb1b4d1e23d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.187ex; height:2.843ex;" alt="{\displaystyle (R-L)/2^{n},}"></span> gdzie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> jest długością ciągu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }"></span> oraz dla dowolnych dwóch <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon ',}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ϵ</mi> <mo>′</mo> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon ',}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9cad3922621d18ac5bf32c537883b282bffa54d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.276ex; height:2.843ex;" alt="{\displaystyle \epsilon ',}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon ''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ϵ</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon ''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1845d57257eaf6afc13587351d09544e2c328931" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.081ex; height:2.509ex;" alt="{\displaystyle \epsilon ''}"></span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{\epsilon ''}\subseteq \Delta _{\epsilon '}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ϵ</mi> <mo>″</mo> </msup> </mrow> </msub> <mo>⊆</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ϵ</mi> <mo>′</mo> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{\epsilon ''}\subseteq \Delta _{\epsilon '}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25d67b3058460f6b3184cd07b2b6c1c503444d42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.2ex; height:2.509ex;" alt="{\displaystyle \Delta _{\epsilon ''}\subseteq \Delta _{\epsilon '}}"></span></dd></dl> <p>wtedy i tylko wtedy, gdy ciąg <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ϵ</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon '}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78f2a815df98759c8f75e757592569d942250171" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.629ex; height:2.509ex;" alt="{\displaystyle \epsilon '}"></span> jest początkiem ciągu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon ''.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ϵ</mi> <mo>″</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon ''.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d4b987a48f00c53bdbc571ea58aa1def2d6e65e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.728ex; height:2.509ex;" alt="{\displaystyle \epsilon ''.}"></span> </p><p>Łatwo też widać, że istnieje nieskończony rosnący ciąg indeksów <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\epsilon _{n})_{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\epsilon _{n})_{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e713c701e614385b097460dc6eaea5b9a07595" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.837ex; height:2.843ex;" alt="{\displaystyle (\epsilon _{n})_{n},}"></span> dla którego każdy z przedziałów <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\Delta }}_{n}=\Delta _{\epsilon _{n}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {\Delta }}_{n}=\Delta _{\epsilon _{n}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05262d781e2214bffde4971f71c11b915cafbedb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.7ex; height:3.176ex;" alt="{\displaystyle {\tilde {\Delta }}_{n}=\Delta _{\epsilon _{n}},}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aedc99ee9cd35f9c34b0f743f8c26b34359ca2fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.299ex; height:2.176ex;" alt="{\displaystyle n\in N}"></span> zawiera nieskończenie wiele wyrazów ciągu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c_{n})_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c_{n})_{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b46f2a186516a05f2e3b4223790184d99385ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.9ex; height:2.843ex;" alt="{\displaystyle (c_{n})_{n}.}"></span> </p><p>Niech teraz <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{0}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{0}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9889aa38e1fa54d522137594d424a2cd95fb1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.71ex; height:2.509ex;" alt="{\displaystyle n_{0}=0}"></span> oraz <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{k+1}=\min\{m>n_{k}:c_{m}\in {\tilde {\Delta }}_{k}\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo movablelimits="true" form="prefix">min</mo> <mo fence="false" stretchy="false">{</mo> <mi>m</mi> <mo>></mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{k+1}=\min\{m>n_{k}:c_{m}\in {\tilde {\Delta }}_{k}\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ebeabbb62bdfce4411cf5b67b8571ac5fc8d51e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.636ex; height:3.176ex;" alt="{\displaystyle n_{k+1}=\min\{m>n_{k}:c_{m}\in {\tilde {\Delta }}_{k}\}.}"></span> Wówczas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n_{k})_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n_{k})_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e01ea2b1febe65f3f2ff008615bb4b92cfcf6e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.381ex; height:2.843ex;" alt="{\displaystyle (n_{k})_{k}}"></span> jest ściśle rosnący oraz <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{n_{k}}\in {\tilde {\Delta }}_{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n_{k}}\in {\tilde {\Delta }}_{k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76add454d24667531dc03cdc23b13a94af644645" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.597ex; height:3.343ex;" alt="{\displaystyle c_{n_{k}}\in {\tilde {\Delta }}_{k}.}"></span> </p><p>Pokażemy, że ciąg <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{n_{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n_{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a89d98f60327e4de8a97d7f191fd68a931e86611" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.085ex; height:2.343ex;" alt="{\displaystyle c_{n_{k}}}"></span> jest zbieżny do <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\star }=\sup\{{\tilde {L}}_{n}:n\in N\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msup> <mo>=</mo> <mo movablelimits="true" form="prefix">sup</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>:</mo> <mi>n</mi> <mo>∈</mo> <mi>N</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\star }=\sup\{{\tilde {L}}_{n}:n\in N\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d2a0aca4bd14921f2a5821c80c11c5c07c28997" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.246ex; height:3.176ex;" alt="{\displaystyle L^{\star }=\sup\{{\tilde {L}}_{n}:n\in N\},}"></span> gdzie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {L}}_{n}=L_{\epsilon _{n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {L}}_{n}=L_{\epsilon _{n}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34bca821a053b42a03608bec5c838b8360a804d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.994ex; height:3.176ex;" alt="{\displaystyle {\tilde {L}}_{n}=L_{\epsilon _{n}}.}"></span> </p><p>Niech zatem <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e04ec3670b50384a3ce48aca42e7cc5131a06b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}"></span> i niech <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> będzie takie, że <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/2^{N}<\varepsilon /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> <mo><</mo> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/2^{N}<\varepsilon /2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efe920c17e4cf65a6f3d54fdf4cf52d0b858cd98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.686ex; height:3.176ex;" alt="{\displaystyle 1/2^{N}<\varepsilon /2}"></span> oraz niech <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> będzie takie, że <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\star }-\varepsilon /2<{\tilde {L}}_{K}\leqslant L^{\star }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msup> <mo>−</mo> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo><</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo>⩽</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\star }-\varepsilon /2<{\tilde {L}}_{K}\leqslant L^{\star }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43a4f298c2101d22d89fba8bdf453851deb4bca4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.643ex; height:3.176ex;" alt="{\displaystyle L^{\star }-\varepsilon /2<{\tilde {L}}_{K}\leqslant L^{\star }.}"></span> </p><p>Biorąc teraz <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geqslant \max\{N,K\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>⩾</mo> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <mi>N</mi> <mo>,</mo> <mi>K</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\geqslant \max\{N,K\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d59cf22aa833f8145e821749842d58192b47868c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.124ex; height:2.843ex;" alt="{\displaystyle k\geqslant \max\{N,K\}}"></span> mamy: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |c_{n_{k}}-L^{\star }|<|c_{n_{k}}-{\tilde {L}}_{k}|+|{\tilde {L}}_{k}-L^{\star }|=|c_{n_{k}}-{\tilde {L}}_{k}|+(L^{\star }-{\tilde {L}}_{k})\leqslant 1/2^{k}+(L^{\star }-{\tilde {L}}_{K})<\varepsilon /2+\varepsilon /2=\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mo>−</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msup> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⩽</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msup> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo><</mo> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>+</mo> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>=</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |c_{n_{k}}-L^{\star }|<|c_{n_{k}}-{\tilde {L}}_{k}|+|{\tilde {L}}_{k}-L^{\star }|=|c_{n_{k}}-{\tilde {L}}_{k}|+(L^{\star }-{\tilde {L}}_{k})\leqslant 1/2^{k}+(L^{\star }-{\tilde {L}}_{K})<\varepsilon /2+\varepsilon /2=\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb395630e73c7029c76cbeea271054df1cf3dbb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:98.931ex; height:3.343ex;" alt="{\displaystyle |c_{n_{k}}-L^{\star }|<|c_{n_{k}}-{\tilde {L}}_{k}|+|{\tilde {L}}_{k}-L^{\star }|=|c_{n_{k}}-{\tilde {L}}_{k}|+(L^{\star }-{\tilde {L}}_{k})\leqslant 1/2^{k}+(L^{\star }-{\tilde {L}}_{K})<\varepsilon /2+\varepsilon /2=\varepsilon }"></span></dd></dl> <p>Tym samym wykazaliśmy zbieżność ciągu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c_{n_{k}})_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c_{n_{k}})_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9e0896cea4414b5a2d7b305ebcf6767c93302c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.983ex; height:3.009ex;" alt="{\displaystyle (c_{n_{k}})_{k}}"></span> do <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\star }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\star }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57c77b628413b9c55d336069254841cec1745e49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.284ex; height:2.343ex;" alt="{\displaystyle L^{\star }.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Trzeci">Trzeci</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&veaction=edit&section=5" title="Edytuj sekcję: Trzeci" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&action=edit&section=5" title="Edytuj kod źródłowy sekcji: Trzeci"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Niech <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c_{n})_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c_{n})_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e0887d353c641f5c00f19fde979b290dea884e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.253ex; height:2.843ex;" alt="{\displaystyle (c_{n})_{n}}"></span> będzie takim ciągiem jak do tej pory, niech <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=\inf\{c_{n}:n\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">inf</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>:</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=\inf\{c_{n}:n\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da18234a8dc24eae4aeb46f54389b593ed85abfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.861ex; height:2.843ex;" alt="{\displaystyle L=\inf\{c_{n}:n\},}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=\sup\{c_{n}:n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">sup</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>:</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=\sup\{c_{n}:n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51b9f1077a8d830cc427dd589a388a439d6b0ef8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.246ex; height:2.843ex;" alt="{\displaystyle R=\sup\{c_{n}:n\}}"></span> i niech <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=(R-L)/2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo>−</mo> <mi>L</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d=(R-L)/2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8b9d678ea9aaddcacd9740af3c04fafcb0eb673" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.283ex; height:2.843ex;" alt="{\displaystyle d=(R-L)/2.}"></span> </p><p>Niech dalej <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{0}=(L+R)/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>L</mi> <mo>+</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{0}=(L+R)/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8f0d32fea9b86b7f1db4a494a0f787d7460b612" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.728ex; height:2.843ex;" alt="{\displaystyle M_{0}=(L+R)/2}"></span> oraz niech <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{k+1}=M_{k}-d/2^{k+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{k+1}=M_{k}-d/2^{k+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc77e6bf70067584a527c0c26e70a3fbb50602d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.455ex; height:3.176ex;" alt="{\displaystyle M_{k+1}=M_{k}-d/2^{k+1}}"></span> jeśli zbiór <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{n:M_{k}-d/2^{k}\leqslant c_{n}\leqslant M_{k}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>n</mi> <mo>:</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>⩽</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⩽</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{n:M_{k}-d/2^{k}\leqslant c_{n}\leqslant M_{k}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a8eb45913d9b4eb586f9180051942a3c553c0f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.235ex; height:3.176ex;" alt="{\displaystyle \{n:M_{k}-d/2^{k}\leqslant c_{n}\leqslant M_{k}\}}"></span> jest nieskończony oraz <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{k+1}=M_{k}+d/2^{k+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{k+1}=M_{k}+d/2^{k+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b1527501a2974dab26ee1780e9d2b4e570c7408" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.455ex; height:3.176ex;" alt="{\displaystyle M_{k+1}=M_{k}+d/2^{k+1}}"></span> w przeciwnym wypadku. Wykażemy indukcyjnie, że przedziały <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{k}=[M_{k}-d/2^{k},M_{k}+d/2^{k}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>,</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{k}=[M_{k}-d/2^{k},M_{k}+d/2^{k}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6923c7ee5904c4177ee2f2031c7aba1533db2103" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.076ex; height:3.176ex;" alt="{\displaystyle \Delta _{k}=[M_{k}-d/2^{k},M_{k}+d/2^{k}]}"></span> zawierają nieskończenie wiele elementów ciągu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c_{n})_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c_{n})_{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b46f2a186516a05f2e3b4223790184d99385ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.9ex; height:2.843ex;" alt="{\displaystyle (c_{n})_{n}.}"></span> </p><p>Ponieważ dla <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa0825f43b484e7021f0d28d2e0d6b94091a1e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.119ex; height:2.509ex;" alt="{\displaystyle k=0,}"></span> mamy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{0}=[M_{0}-d,M_{0}+d]=[L,R],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>d</mi> <mo>,</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>d</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mi>L</mi> <mo>,</mo> <mi>R</mi> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{0}=[M_{0}-d,M_{0}+d]=[L,R],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/496513937b7bfcfd69eca1b7e02aed058215b085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.565ex; height:2.843ex;" alt="{\displaystyle \Delta _{0}=[M_{0}-d,M_{0}+d]=[L,R],}"></span> baza indukcji jest prawdziwa. </p><p>Załóżmy zatem, że dla pewnego <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> przedział <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{k}=[M_{k}-d/2^{k},M_{k}+d/2^{k}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>,</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{k}=[M_{k}-d/2^{k},M_{k}+d/2^{k}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6923c7ee5904c4177ee2f2031c7aba1533db2103" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.076ex; height:3.176ex;" alt="{\displaystyle \Delta _{k}=[M_{k}-d/2^{k},M_{k}+d/2^{k}]}"></span> zawiera nieskończenie wiele elementów ciągu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c_{n})_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c_{n})_{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b46f2a186516a05f2e3b4223790184d99385ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.9ex; height:2.843ex;" alt="{\displaystyle (c_{n})_{n}.}"></span> Jeśli zbiór <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{n:M_{k}-d/2^{k}\leqslant c_{n}\leqslant M_{k}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>n</mi> <mo>:</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>⩽</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⩽</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{n:M_{k}-d/2^{k}\leqslant c_{n}\leqslant M_{k}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a8eb45913d9b4eb586f9180051942a3c553c0f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.235ex; height:3.176ex;" alt="{\displaystyle \{n:M_{k}-d/2^{k}\leqslant c_{n}\leqslant M_{k}\}}"></span> jest nieskończony, to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{k+1}=M_{k}-d/2^{k+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{k+1}=M_{k}-d/2^{k+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc77e6bf70067584a527c0c26e70a3fbb50602d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.455ex; height:3.176ex;" alt="{\displaystyle M_{k+1}=M_{k}-d/2^{k+1}}"></span> i wówczas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{k+1}=[M_{k+1}-d/2^{k+1},M_{k+1}+d/2^{k+1}]=[M_{k}-d/2^{k},M_{k}],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>,</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{k+1}=[M_{k+1}-d/2^{k+1},M_{k+1}+d/2^{k+1}]=[M_{k}-d/2^{k},M_{k}],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7e646be8d7d73d5db5088c5641761eeb83094a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:60.807ex; height:3.176ex;" alt="{\displaystyle \Delta _{k+1}=[M_{k+1}-d/2^{k+1},M_{k+1}+d/2^{k+1}]=[M_{k}-d/2^{k},M_{k}],}"></span> czyli zawiera nieskończenie wiele elementów rozważanego ciągu. </p><p>Jeśli zbiór <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{n:M_{k}-d/2^{k}\leqslant c_{n}\leqslant M_{k}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>n</mi> <mo>:</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>⩽</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⩽</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{n:M_{k}-d/2^{k}\leqslant c_{n}\leqslant M_{k}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a8eb45913d9b4eb586f9180051942a3c553c0f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.235ex; height:3.176ex;" alt="{\displaystyle \{n:M_{k}-d/2^{k}\leqslant c_{n}\leqslant M_{k}\}}"></span> nieskończony nie jest, to musi być nieskończony <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{n:M_{k}\leqslant c_{n}\leqslant M_{k}+d/2^{k}\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>n</mi> <mo>:</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⩽</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⩽</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{n:M_{k}\leqslant c_{n}\leqslant M_{k}+d/2^{k}\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2270fcf94d97c34e51fbb7f2c63f8f9891d8cee7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.881ex; height:3.176ex;" alt="{\displaystyle \{n:M_{k}\leqslant c_{n}\leqslant M_{k}+d/2^{k}\},}"></span> na mocy założenia indukcyjnego i wówczas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{k+1}=M_{k}+d/2^{k+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{k+1}=M_{k}+d/2^{k+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b1527501a2974dab26ee1780e9d2b4e570c7408" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.455ex; height:3.176ex;" alt="{\displaystyle M_{k+1}=M_{k}+d/2^{k+1}}"></span> oraz <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{k+1}=[M_{k+1}-d/2^{k+1},M_{k+1}+d/2^{k+1}]=[M_{k},M_{k}+d/2^{k}],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{k+1}=[M_{k+1}-d/2^{k+1},M_{k+1}+d/2^{k+1}]=[M_{k},M_{k}+d/2^{k}],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cf1b02246fd64c2f6f8104828a91c70cd942a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:60.807ex; height:3.176ex;" alt="{\displaystyle \Delta _{k+1}=[M_{k+1}-d/2^{k+1},M_{k+1}+d/2^{k+1}]=[M_{k},M_{k}+d/2^{k}],}"></span> co dopełnia dowód kroku indukcyjnego. </p><p>Niech teraz <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{0}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{0}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/225a6f46051451ffb75d945be4a60e4a441a8682" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.356ex; height:2.509ex;" alt="{\displaystyle m_{0}=0}"></span> i niech <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{k+1}=\min\{n>m_{k}:c_{n}\in \Delta _{k+1}\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo movablelimits="true" form="prefix">min</mo> <mo fence="false" stretchy="false">{</mo> <mi>n</mi> <mo>></mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{k+1}=\min\{n>m_{k}:c_{n}\in \Delta _{k+1}\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97f3ab74f6854ca65f1972f879c7969e546fe7b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.926ex; height:2.843ex;" alt="{\displaystyle m_{k+1}=\min\{n>m_{k}:c_{n}\in \Delta _{k+1}\}.}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c_{m_{n}})_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c_{m_{n}})_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ef1a7be7542975a97998724b0697cc66fbf2560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.675ex; height:2.843ex;" alt="{\displaystyle (c_{m_{n}})_{n}}"></span> jest podciągiem ciągu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c_{n})_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c_{n})_{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b46f2a186516a05f2e3b4223790184d99385ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.9ex; height:2.843ex;" alt="{\displaystyle (c_{n})_{n}.}"></span> Ciąg <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{n}=M_{n}-d/2^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{n}=M_{n}-d/2^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c451e621f940fb49f4723818cd4ce03783121741" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.972ex; height:2.843ex;" alt="{\displaystyle L_{n}=M_{n}-d/2^{n}}"></span> jest rosnący i ograniczony, więc posiada supremum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\star }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\star }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57c77b628413b9c55d336069254841cec1745e49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.284ex; height:2.343ex;" alt="{\displaystyle L^{\star }.}"></span> Pokażemy, że <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\star }=\lim _{n\to \infty }c_{m_{n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msup> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\star }=\lim _{n\to \infty }c_{m_{n}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90ac20d88f5dea517c7a6564bdff4777a32f0858" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.689ex; height:3.843ex;" alt="{\displaystyle L^{\star }=\lim _{n\to \infty }c_{m_{n}}.}"></span> </p><p>Niech w tym celu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e04ec3670b50384a3ce48aca42e7cc5131a06b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}"></span> i niech <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> będzie takie, że <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/2^{N}<\varepsilon /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> <mo><</mo> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/2^{N}<\varepsilon /2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efe920c17e4cf65a6f3d54fdf4cf52d0b858cd98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.686ex; height:3.176ex;" alt="{\displaystyle 1/2^{N}<\varepsilon /2}"></span> oraz niech <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> będzie takie, że <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\star }-\varepsilon /2<{L}_{K}\leqslant L^{\star }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msup> <mo>−</mo> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo><</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo>⩽</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\star }-\varepsilon /2<{L}_{K}\leqslant L^{\star }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3f81eb1d8a34af5477cea5b0b3a9d0271c76d0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.643ex; height:2.843ex;" alt="{\displaystyle L^{\star }-\varepsilon /2<{L}_{K}\leqslant L^{\star }.}"></span> </p><p>Biorąc teraz <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geqslant \max\{N,K\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>⩾</mo> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <mi>N</mi> <mo>,</mo> <mi>K</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geqslant \max\{N,K\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1788bdb88fff519381c875482fddb8a86c8d0383" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.307ex; height:2.843ex;" alt="{\displaystyle n\geqslant \max\{N,K\}}"></span> mamy: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |c_{m_{n}}-L^{\star }|<|c_{m_{n}}-{L}_{n}|+|{L}_{n}-L^{\star }|=(c_{m_{n}}-{L}_{n})+(L^{\star }-{L}_{n})\leqslant 1/2^{n}+(L^{\star }-{L}_{K})<\varepsilon /2+\varepsilon /2=\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo>−</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msup> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⩽</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msup> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo><</mo> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>+</mo> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>=</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |c_{m_{n}}-L^{\star }|<|c_{m_{n}}-{L}_{n}|+|{L}_{n}-L^{\star }|=(c_{m_{n}}-{L}_{n})+(L^{\star }-{L}_{n})\leqslant 1/2^{n}+(L^{\star }-{L}_{K})<\varepsilon /2+\varepsilon /2=\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dcc951d6d1defd33237149f7e53e8849c6f52cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:101.781ex; height:2.843ex;" alt="{\displaystyle |c_{m_{n}}-L^{\star }|<|c_{m_{n}}-{L}_{n}|+|{L}_{n}-L^{\star }|=(c_{m_{n}}-{L}_{n})+(L^{\star }-{L}_{n})\leqslant 1/2^{n}+(L^{\star }-{L}_{K})<\varepsilon /2+\varepsilon /2=\varepsilon }"></span></dd></dl> <p>Tym samym wykazaliśmy zbieżność ciągu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c_{m_{n}})_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c_{m_{n}})_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ef1a7be7542975a97998724b0697cc66fbf2560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.675ex; height:2.843ex;" alt="{\displaystyle (c_{m_{n}})_{n}}"></span> do <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\star }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\star }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57c77b628413b9c55d336069254841cec1745e49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.284ex; height:2.343ex;" alt="{\displaystyle L^{\star }.}"></span> </p><p>Zauważmy, że <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\star }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\star }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63534a684ef247f73d6c3ca205b58ec097de82a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.343ex;" alt="{\displaystyle L^{\star }}"></span> jest także granicą ciągów <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M_{n})_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M_{n})_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12d10f871b9500523c394f2de8d267120993c29e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.5ex; height:2.843ex;" alt="{\displaystyle (M_{n})_{n}}"></span> oraz <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{n}=M_{n}+d/2^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{n}=M_{n}+d/2^{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dda88ead555b3638510a0f5934ca560555bc2111" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.8ex; height:2.843ex;" alt="{\displaystyle R_{n}=M_{n}+d/2^{n}.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Uwagi">Uwagi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&veaction=edit&section=6" title="Edytuj sekcję: Uwagi" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&action=edit&section=6" title="Edytuj kod źródłowy sekcji: Uwagi"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="do-not-make-smaller refsection refsection-uwagi ll-script ll-script-uwagi"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">W literaturze niemal wyłącznie występuje błędna tj. nieodmieniona forma pierwszego nazwiska: Twierdzenie Bolzano-Weierstrassa.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Literatura_dodatkowa">Literatura dodatkowa</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&veaction=edit&section=7" title="Edytuj sekcję: Literatura dodatkowa" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&action=edit&section=7" title="Edytuj kod źródłowy sekcji: Literatura dodatkowa"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation book"><a href="/wiki/Grigorij_Fichtenholz" title="Grigorij Fichtenholz">Grigorij Fichtenholz</a>: <i>Rachunek różniczkowy i całkowy</i>. Warszawa: <a href="/wiki/Wydawnictwo_Naukowe_PWN" title="Wydawnictwo Naukowe PWN">Państwowe Wydawnictwo Naukowe</a>, 1972.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Rachunek+r%C3%B3%C5%BCniczkowy+i+ca%C5%82kowy&rft.au=%5B%5BGrigorij+Fichtenholz%5D%5D&rft.date=1972&rft.pub=%5B%5BWydawnictwo+Naukowe+PWN%7CPa%C5%84stwowe+Wydawnictwo+Naukowe%5D%5D&rft.place=Warszawa"></span></cite><span class="problemy" aria-hidden="true" data-nosnippet=""> Brak numerów stron w książce</span></li> <li><cite class="citation book"><a href="/wiki/Franciszek_Leja" title="Franciszek Leja">Franciszek Leja</a>: <i>Rachunek różniczkowy i całkowy</i>. Warszawa: PWN, 1973.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Rachunek+r%C3%B3%C5%BCniczkowy+i+ca%C5%82kowy&rft.au=%5B%5BFranciszek+Leja%5D%5D&rft.date=1973&rft.pub=PWN&rft.place=Warszawa"></span></cite><span class="problemy" aria-hidden="true" data-nosnippet=""> Brak numerów stron w książce</span></li> <li><cite class="citation book"><a href="/wiki/Walter_Rudin" title="Walter Rudin">Walter Rudin</a>: <i>Podstawy analizy matematycznej</i>. Warszawa: PWN, 1976.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Podstawy+analizy+matematycznej&rft.au=%5B%5BWalter+Rudin%5D%5D&rft.date=1976&rft.pub=PWN&rft.place=Warszawa"></span></cite><span class="problemy" aria-hidden="true" data-nosnippet=""> Brak numerów stron w książce</span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Linki_zewnętrzne"><span id="Linki_zewn.C4.99trzne"></span>Linki zewnętrzne</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&veaction=edit&section=8" title="Edytuj sekcję: Linki zewnętrzne" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&action=edit&section=8" title="Edytuj kod źródłowy sekcji: Linki zewnętrzne"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span typeof="mw:File"><a href="/wiki/Otwarty_dost%C4%99p" title="publikacja w otwartym dostępie – możesz ją przeczytać"><img alt="publikacja w otwartym dostępie – możesz ją przeczytać" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Open_Access_logo_green_alt2.svg/8px-Open_Access_logo_green_alt2.svg.png" decoding="async" width="8" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Open_Access_logo_green_alt2.svg/12px-Open_Access_logo_green_alt2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/91/Open_Access_logo_green_alt2.svg/16px-Open_Access_logo_green_alt2.svg.png 2x" data-file-width="640" data-file-height="1000" /></a></span> <i><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Bolzano-Weierstrass_theorem">Bolzano-Weierstrass theorem</a></i>, Encyclopedia of Mathematics, encyclopediaofmath.org [dostęp 2024-07-09].</li> <li><cite class="citation open-access"><span class="cite-name-before"><span class="cite-name-full">Eric W.</span><span class="cite-name-initials" title="Eric W." style="display:none">E.W.</span> </span><span class="cite-lastname">Weisstein</span><span class="cite-name-after" style="display:none"> <span class="cite-name-full">Eric W.</span><span class="cite-name-initials" title="Eric W.">E.W.</span></span>, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Bolzano-WeierstrassTheorem.html"><i>Bolzano-Weierstrass Theorem</i></a>, [w:] <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>, <a href="/wiki/Wolfram_Research" title="Wolfram Research">Wolfram Research</a><span class="Z3988" title="ctx_ver=Z39.88-2004&rft.gengre=bookitem&rft.aufirst=Eric+W.&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.btitle=%5B%5BMathWorld%5D%5D&rft.atitle=Bolzano-Weierstrass+Theorem&rft.aulast=Weisstein&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FBolzano-WeierstrassTheorem.html" style="display:none"> </span> <span class="lang-list">(<abbr title="Treść w języku angielskim (English)">ang.</abbr>)</span>.</cite> [dostęp 2022-10-09].</li></ul> <div class="navbox do-not-make-smaller mw-collapsible mw-collapsed" data-expandtext="pokaż" data-collapsetext="ukryj"><style data-mw-deduplicate="TemplateStyles:r74983602">.mw-parser-output .navbox{border:1px solid var(--border-color-base,#a2a9b1);margin:auto;text-align:center;padding:3px;margin-top:1em;clear:both}.mw-parser-output table.navbox:not(.pionowy){width:100%}.mw-parser-output .navbox+.navbox{border-top:0;margin-top:0}.mw-parser-output .navbox.pionowy{width:250px;float:right;clear:right;margin:0 0 0.4em 1.4em}.mw-parser-output .navbox.pionowy .before,.mw-parser-output .navbox.pionowy .after{padding:0.5em 0;text-align:center}.mw-parser-output .navbox>.caption,.mw-parser-output .navbox>tbody>tr>th{background:#ccf;text-align:center;font-weight:bold}.mw-parser-output .navbox .tnavbar{font-weight:normal;font-size:xx-small;white-space:nowrap;padding:0}.mw-parser-output .navbox>.tnavbar{margin-left:1em;float:left}.mw-parser-output .navbox .below>hr+.tnavbar{margin-left:auto;margin-right:auto}.mw-parser-output .navbox .below>.tnavbar:before{content:"Ten szablon: "}.mw-parser-output .navbox .tnavbar li:after{content:" · "}.mw-parser-output .navbox .tnavbar li:last-child:after{content:none}.mw-parser-output .navbox hr{margin:0.2em 1em}.mw-parser-output .navbox .title{background:#ddf;text-align:center;font-weight:bold}.mw-parser-output .navbox>.mw-collapsible-content:not(.grupa-szablonów-nawigacyjnych){margin-top:2px;padding:0;font-size:smaller;overflow:auto}.mw-parser-output .navbox .above+div,.mw-parser-output .navbox .above+.navbox-main-content,.mw-parser-output .navbox .below,.mw-parser-output .navbox .title+.grid{margin-top:2px}.mw-parser-output .navbox>.mw-collapsible-content>.above,.mw-parser-output .navbox>.mw-collapsible-content>.below{background:#ddf;text-align:center;margin-left:auto;margin-right:auto}.mw-parser-output .navbox:not(.pionowy) .flex{display:flex;flex-direction:row}.mw-parser-output .navbox .flex>.before,.mw-parser-output .navbox .flex>.after{align-self:center;text-align:center}.mw-parser-output .navbox .flex>.navbox-main-content{flex-grow:1}.mw-parser-output .navbox:not(.pionowy) .before{margin-right:0.5em}.mw-parser-output .navbox:not(.pionowy) .after{margin-left:0.5em}.mw-parser-output .navbox .inner-columns,.mw-parser-output .navbox .inner-group,.mw-parser-output .navbox .inner-standard{border-spacing:0;border-collapse:collapse;width:100%}.mw-parser-output .navbox .inner-standard>tbody>tr>.opis{text-align:right;vertical-align:middle}.mw-parser-output .navbox .inner-standard>tbody>tr>.opis+.spis{border-left:2px solid var(--background-color-base,#fff);text-align:left}.mw-parser-output .navbox .inner-standard>tbody>tr>td{padding:0;width:100%}.mw-parser-output .navbox .inner-standard>tbody>tr>td:first-child{text-align:center}.mw-parser-output .navbox .inner-standard .inner-standard>tbody>tr>td{text-align:left}.mw-parser-output .navbox .inner-standard>tbody>tr>.navbox-odd,.mw-parser-output .navbox .inner-standard>tbody>tr>.navbox-even{padding:0 0.3em}.mw-parser-output .navbox .inner-standard>tbody>tr+tr>th,.mw-parser-output .navbox .inner-standard>tbody>tr+tr>td{border-top:2px solid var(--background-color-base,#fff)}.mw-parser-output .navbox .inner-standard>tbody>tr>th+td{border-left:2px solid var(--background-color-base,#fff)}.mw-parser-output .navbox .inner-columns{table-layout:fixed}.mw-parser-output .navbox .inner-columns>tbody>tr>th,.mw-parser-output .navbox .inner-columns>tbody>tr>td{padding:0;border-left:2px solid var(--background-color-base,#fff);border-right:2px solid var(--background-color-base,#fff)}.mw-parser-output .navbox .inner-columns>tbody>tr>td{vertical-align:top}.mw-parser-output .navbox .inner-columns>tbody>tr+tr>td{border-top:2px solid var(--background-color-base,#fff)}.mw-parser-output .navbox .inner-columns>tbody>tr>th:first-child,.mw-parser-output .navbox .inner-columns>tbody>tr>td:first-child{border-left:0}.mw-parser-output .navbox .inner-columns>tbody>tr>th:last-child,.mw-parser-output .navbox .inner-columns>tbody>tr>td:last-child{border-right:0}.mw-parser-output .navbox .inner-columns>tbody>tr>td>ul,.mw-parser-output .navbox .inner-columns>tbody>tr>td>ol,.mw-parser-output .navbox .inner-columns>tbody>tr>td>dl{text-align:left;column-width:24em}.mw-parser-output .navbox .inner-group>div+div,.mw-parser-output .navbox .inner-group>div>div+div,.mw-parser-output .navbox .inner-group>div>div+table{margin-top:2px}.mw-parser-output .navbox .inner-group>div>.opis,.mw-parser-output .navbox .inner-group>div>.spis{padding:0.1em 1em;text-align:center}.mw-parser-output .navbox>.mw-collapsible-toggle,.mw-parser-output .navbox .inner-group>div.mw-collapsible>.mw-collapsible-toggle{width:4em;text-align:right;margin-right:0.4em}.mw-parser-output .navbox>.fakebar,.mw-parser-output .navbox .inner-group>div.mw-collapsible>.fakebar{float:left;width:4em;height:1em}.mw-parser-output .navbox .opis{background:#ddf;padding:0 1em;white-space:nowrap;font-weight:bold}.mw-parser-output .navbox.pionowy .opis{white-space:normal}.mw-parser-output .navbox.pionowy .navbox-even,.mw-parser-output .navbox:not(.pionowy) .navbox-odd{background:transparent}.mw-parser-output .navbox.pionowy .navbox-odd,.mw-parser-output .navbox:not(.pionowy) .navbox-even{background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .navbox .inner-group>div>div+div{background:transparent}.mw-parser-output .navbox p{margin:0;padding:0.3em 0}.mw-parser-output .navbox.medaliści .opis.a1,.mw-parser-output .navbox.medaliści .a1 .opis{background:gold}.mw-parser-output .navbox.medaliści .opis.a2,.mw-parser-output .navbox.medaliści .a2 .opis{background:silver}.mw-parser-output .navbox.medaliści .opis.a3,.mw-parser-output .navbox.medaliści .a3 .opis{background:#c96}.mw-parser-output .navbox .navbox-main-content>ul,.mw-parser-output .navbox .navbox-main-content>dl,.mw-parser-output .navbox .navbox-main-content>ol{column-width:24em;text-align:left}.mw-parser-output .navbox ul{list-style:none}.mw-parser-output .navbox .references{background:transparent}.mw-parser-output .navbox .hwrap .hlist dd,.mw-parser-output .navbox .hwrap .hlist dt,.mw-parser-output .navbox .hwrap .hlist li{white-space:normal}.mw-parser-output .navbox .rok{display:inline-block;width:4em;padding-right:0.5em;text-align:right}.mw-parser-output .navbox .navbox-statistics{margin-top:2px;border-top:1px solid var(--border-color-base,#a2a9b1);text-align:center;font-size:small}.mw-parser-output .navbox-summary>.title{font-weight:bold;font-size:larger}.mw-parser-output .navbox:not(.grupa-szablonów) .navbox{margin:0;border:0;padding:0}.mw-parser-output .navbox.grupa-szablonów>.grupa-szablonów-nawigacyjnych>.navbox:first-child{margin-top:2px}@media(max-width:800px){.mw-parser-output .navbox:not(.pionowy) .flex>.before,.mw-parser-output .navbox:not(.pionowy) .flex>.after{display:none}}.mw-parser-output .navbox .opis img,.mw-parser-output .navbox .opis .flagicon,.mw-parser-output .navbox>.caption>.flagicon,.mw-parser-output .navbox>.caption>.image{display:none}@media screen{html.skin-theme-clientpref-night .mw-parser-output .navbox>.caption,html.skin-theme-clientpref-night .mw-parser-output .navbox>tbody>tr>th{background-color:#3a3c3e}html.skin-theme-clientpref-night .mw-parser-output .navbox .title,html.skin-theme-clientpref-night .mw-parser-output .navbox>.mw-collapsible-content>.above,html.skin-theme-clientpref-night .mw-parser-output .navbox>.mw-collapsible-content>.below,html.skin-theme-clientpref-night .mw-parser-output .navbox .opis{background-color:#303234}html.skin-theme-clientpref-night .mw-parser-output .navbox.medaliści .opis.a1,html.skin-theme-clientpref-night .mw-parser-output .navbox.medaliści .a1 .opis{background:#715f00}html.skin-theme-clientpref-night .mw-parser-output .navbox.medaliści .opis.a2,html.skin-theme-clientpref-night .mw-parser-output .navbox.medaliści .a2 .opis{background:#5f5f5f}html.skin-theme-clientpref-night .mw-parser-output .navbox.medaliści .opis.a3,html.skin-theme-clientpref-night .mw-parser-output .navbox.medaliści .a3 .opis{background:#764617}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbox>.caption,html.skin-theme-clientpref-os .mw-parser-output .navbox>tbody>tr>th{background-color:#3a3c3e}html.skin-theme-clientpref-os .mw-parser-output .navbox .title,html.skin-theme-clientpref-os .mw-parser-output .navbox>.mw-collapsible-content>.above,html.skin-theme-clientpref-os .mw-parser-output .navbox>.mw-collapsible-content>.below,html.skin-theme-clientpref-os .mw-parser-output .navbox .opis{background-color:#303234}html.skin-theme-clientpref-os .mw-parser-output .navbox.medaliści .opis.a1,html.skin-theme-clientpref-os .mw-parser-output .navbox.medaliści .a1 .opis{background:#715f00}html.skin-theme-clientpref-os .mw-parser-output .navbox.medaliści .opis.a2,html.skin-theme-clientpref-os .mw-parser-output .navbox.medaliści .a2 .opis{background:#5f5f5f}html.skin-theme-clientpref-os .mw-parser-output .navbox.medaliści .opis.a3,html.skin-theme-clientpref-os .mw-parser-output .navbox.medaliści .a3 .opis{background:#764617}}</style><ul class="tnavbar noprint plainlinks hlist"><li><a href="/wiki/Szablon:Ci%C4%85gi_liczbowe" title="Szablon:Ciągi liczbowe"><span title="Pokaż ten szablon">p</span></a></li><li><a href="/w/index.php?title=Dyskusja_szablonu:Ci%C4%85gi_liczbowe&action=edit&redlink=1" class="new" title="Dyskusja szablonu:Ciągi liczbowe (strona nie istnieje)"><span title="Dyskusja na temat tego szablonu">d</span></a></li><li title="Możesz edytować ten szablon. Użyj przycisku podglądu przed zapisaniem zmian."><a class="external text" href="https://pl.wikipedia.org/w/index.php?title=Szablon:Ci%C4%85gi_liczbowe&action=edit">e</a></li></ul><div class="navbox-title caption"><a href="/wiki/Ci%C4%85g_(matematyka)" title="Ciąg (matematyka)">Ciągi</a> liczbowe</div><div class="mw-collapsible-content"><table class="navbox-main-content inner-standard"><tbody><tr class="a1"><th class="navbox-group opis" scope="row">pojęcia<br />definiujące</th><td class="navbox-list spis"><table class="inner-standard"><tbody><tr class="a1_1"><th class="navbox-group opis" scope="row">ciągi ogólne</th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Funkcja" title="Funkcja">funkcja</a></li> <li><a href="/wiki/Dziedzina_(matematyka)" title="Dziedzina (matematyka)">dziedzina</a></li> <li><a href="/wiki/Liczby_naturalne" title="Liczby naturalne">liczby naturalne</a></li> <li><a href="/wiki/Podzbi%C3%B3r" title="Podzbiór">podzbiór</a></li></ul> </td></tr><tr class="a1_2"><th class="navbox-group opis" scope="row">ciągi liczbowe</th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Przeciwdziedzina" title="Przeciwdziedzina">przeciwdziedzina</a></li> <li><a href="/wiki/Liczba" title="Liczba">liczba</a></li></ul> </td></tr></tbody></table></td></tr><tr class="a2"><th class="navbox-group opis" scope="row">typy ciągów</th><td class="navbox-list spis"><table class="inner-standard"><tbody><tr class="a2_1"><th class="navbox-group opis" scope="row">skończone</th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Para_uporz%C4%85dkowana" title="Para uporządkowana">para uporządkowana</a></li> <li><a href="/wiki/Krotka_(struktura_danych)" title="Krotka (struktura danych)">krotka</a></li> <li><a href="/wiki/Lista" title="Lista">lista</a></li></ul> </td></tr><tr class="a2_2"><th class="navbox-group opis" scope="row"><a href="/wiki/Niesko%C5%84czono%C5%9B%C4%87" title="Nieskończoność">nieskończone</a></th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Funkcja_ograniczona" title="Funkcja ograniczona">nieograniczone</a></li> <li><a href="/wiki/Ci%C4%85g_Cauchy%E2%80%99ego" title="Ciąg Cauchy’ego">Cauchy’ego</a></li> <li><a href="/wiki/Granica_ci%C4%85gu" title="Granica ciągu">zbieżne i rozbieżne</a> <ul><li><a href="/wiki/Granica_niew%C5%82a%C5%9Bciwa_funkcji" title="Granica niewłaściwa funkcji">rozbieżne do nieskończoności</a></li></ul></li> <li><a href="/wiki/Sumowalno%C5%9B%C4%87_metod%C4%85_Ces%C3%A0ro" title="Sumowalność metodą Cesàro">sumowalne metodą Cesàro</a></li> <li><a href="/wiki/Szereg_(matematyka)" title="Szereg (matematyka)">szeregi liczbowe</a> <ul><li><a href="/wiki/Szereg_geometryczny" title="Szereg geometryczny">geometryczne</a></li> <li><a href="/wiki/Szereg_harmoniczny" title="Szereg harmoniczny">harmoniczne</a></li> <li><a href="/wiki/Szereg_naprzemienny" title="Szereg naprzemienny">naprzemienne</a></li> <li><a href="/wiki/Szereg_Dirichleta" title="Szereg Dirichleta">Dirichleta</a></li> <li><a href="/wiki/U%C5%82amek_dziesi%C4%99tny_niesko%C5%84czony" title="Ułamek dziesiętny nieskończony">ułamki dziesiętne nieskończone</a></li></ul></li> <li><a href="/wiki/Iloczyn_niesko%C5%84czony" title="Iloczyn nieskończony">nieskończone iloczyny</a> liczbowe</li> <li><a href="/wiki/U%C5%82amek_%C5%82a%C5%84cuchowy" title="Ułamek łańcuchowy">ułamki łańcuchowe</a></li> <li><a href="/wiki/Funkcja_arytmetyczna" title="Funkcja arytmetyczna">funkcje arytmetyczne</a> <ul><li><a href="/wiki/Funkcja_addytywna_(teoria_liczb)" title="Funkcja addytywna (teoria liczb)">addytywne</a></li> <li><a href="/wiki/Funkcja_multiplikatywna" title="Funkcja multiplikatywna">multyplikatywne</a></li></ul></li></ul> </td></tr><tr class="a2_3"><th class="navbox-group opis" scope="row"><a href="/wiki/Funkcja_monotoniczna" title="Funkcja monotoniczna">monotoniczne</a></th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Funkcja_monotoniczna" title="Funkcja monotoniczna">niemalejące</a></li> <li><a href="/wiki/Funkcja_monotoniczna" title="Funkcja monotoniczna">nierosnące</a></li> <li><a href="/wiki/Funkcja_monotoniczna" title="Funkcja monotoniczna">rosnące</a> <ul><li><a href="/wiki/Ci%C4%85g_superrosn%C4%85cy" title="Ciąg superrosnący">superrosnące</a></li></ul></li> <li><a href="/wiki/Funkcja_monotoniczna" title="Funkcja monotoniczna">malejące</a></li> <li><a href="/wiki/Funkcja_sta%C5%82a" title="Funkcja stała">stałe</a></li></ul> </td></tr><tr class="a2_4"><th class="navbox-group opis" scope="row">inne</th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Funkcja_r%C3%B3%C5%BCnowarto%C5%9Bciowa" title="Funkcja różnowartościowa">różnowartościowe</a> <ul><li><a href="/wiki/Permutacja" title="Permutacja">permutacje</a></li></ul></li> <li><a href="/wiki/Funkcja_ograniczona" title="Funkcja ograniczona">ograniczone</a> <ul><li><a href="/wiki/Funkcja_okresowa" title="Funkcja okresowa">okresowe</a></li></ul></li> <li><a href="/wiki/Ci%C4%85g_arytmetyczny" title="Ciąg arytmetyczny">arytmetyczne</a></li> <li><a href="/wiki/Ci%C4%85g_geometryczny" title="Ciąg geometryczny">geometryczne</a></li> <li><a href="/wiki/U%C5%82amek_dziesi%C4%99tny" title="Ułamek dziesiętny">ułamki dziesiętne</a></li></ul> </td></tr></tbody></table></td></tr><tr class="a3"><th class="navbox-group opis" scope="row">przykłady ciągów<br />liczb naturalnych</th><td class="navbox-list spis"><table class="inner-standard"><tbody><tr class="a3_1"><th class="navbox-group opis" scope="row">niemalejące</th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Ci%C4%85g_Eulera" title="Ciąg Eulera">ciąg Eulera</a></li> <li><a href="/wiki/Ci%C4%85g_Fibonacciego" title="Ciąg Fibonacciego">ciąg Fibonacciego</a></li> <li><a href="/wiki/Funkcja_licz%C4%85ca_liczby_pierwsze" title="Funkcja licząca liczby pierwsze">funkcja pi (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> – licząca liczby pierwsze)</a></li> <li><a href="/wiki/Funkcja_Liouville%E2%80%99a" title="Funkcja Liouville’a">funkcja Liouville’a</a></li> <li><a href="/wiki/Liczby_Armstronga" title="Liczby Armstronga">liczby Armstronga</a></li> <li><a href="/wiki/Liczby_autobiograficzne" title="Liczby autobiograficzne">liczby autobiograficzne</a></li> <li><a href="/wiki/Liczby_automorficzne" title="Liczby automorficzne">liczby automorficzne</a></li> <li><a href="/wiki/Liczby_bezkwadratowe" title="Liczby bezkwadratowe">liczby bezkwadratowe</a></li> <li><a href="/wiki/Liczby_Carmichaela" title="Liczby Carmichaela">liczby Carmichaela</a></li> <li><a href="/wiki/Liczby_Cullena" title="Liczby Cullena">liczby Cullena</a></li> <li><a href="/wiki/Liczby_czworo%C5%9Bcienne" title="Liczby czworościenne">liczby czworościenne</a></li> <li><a href="/wiki/Liczby_deficytowe" title="Liczby deficytowe">liczby deficytowe</a></li> <li><a href="/wiki/Liczby_doskona%C5%82e" title="Liczby doskonałe">liczby doskonałe</a></li> <li><a href="/wiki/Liczby_Fermata" title="Liczby Fermata">liczby Fermata</a></li> <li><a href="/wiki/Liczby_g%C5%82adkie" title="Liczby gładkie">liczby gładkie</a></li> <li><a href="/wiki/Liczby_kwadratowe" title="Liczby kwadratowe">liczby kwadratowe</a></li> <li><a href="/wiki/Liczby_Mersenne%E2%80%99a" title="Liczby Mersenne’a">liczby Mersenne’a</a></li> <li><a href="/wiki/Liczby_nadmiarowe" title="Liczby nadmiarowe">liczby nadmiarowe</a></li> <li><a href="/wiki/Liczby_Nivena" title="Liczby Nivena">liczby Nivena</a></li> <li><a href="/wiki/Palindrom" title="Palindrom">liczby palindromiczne</a></li> <li><a href="/wiki/Liczby_Pella" title="Liczby Pella">liczby Pella</a></li> <li><a href="/wiki/Liczby_pierwsze" title="Liczby pierwsze">liczby pierwsze</a></li> <li><a href="/wiki/Liczby_piramidalne" title="Liczby piramidalne">liczby piramidalne</a></li> <li><a href="/wiki/Liczby_p%C3%B3%C5%82pierwsze" title="Liczby półpierwsze">liczby półpierwsze</a></li> <li><a href="/wiki/Liczby_pseudopierwsze" title="Liczby pseudopierwsze">liczby pseudopierwsze</a></li> <li><a href="/wiki/Liczby_Smitha" title="Liczby Smitha">liczby Smitha</a></li> <li><a href="/wiki/Liczby_sfeniczne" title="Liczby sfeniczne">liczby sfeniczne</a></li> <li><a href="/wiki/Liczby_Sierpi%C5%84skiego" title="Liczby Sierpińskiego">liczby Sierpińskiego</a></li> <li><a href="/wiki/Liczby_taks%C3%B3wkowe" title="Liczby taksówkowe">liczby taksówkowe</a></li> <li><a href="/wiki/Liczby_tr%C3%B3jk%C4%85tne" title="Liczby trójkątne">liczby trójkątne</a></li> <li><a href="/wiki/Liczby_weso%C5%82e" title="Liczby wesołe">liczby wesołe</a></li> <li><a href="/wiki/Liczby_z%C5%82o%C5%BCone" title="Liczby złożone">liczby złożone</a></li> <li><a href="/wiki/Silnia" title="Silnia">silnia</a></li></ul> </td></tr><tr class="a3_2"><th class="navbox-group opis" scope="row">inne</th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Problem_Collatza" title="Problem Collatza">funkcja Collatza</a></li> <li><a href="/wiki/Funkcja_%CF%84" title="Funkcja τ">funkcja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> (tau)</a></li> <li><a href="/wiki/Funkcja_%CF%83" title="Funkcja σ">funkcja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></span> (sigma)</a></li> <li><a href="/wiki/Funkcja_M%C3%B6biusa" title="Funkcja Möbiusa">funkcja Möbiusa</a></li> <li><a href="/wiki/Funkcja_Mertensa" title="Funkcja Mertensa">funkcja Mertensa</a></li> <li><a href="/wiki/Funkcja_%CF%86" title="Funkcja φ">tocjent Eulera</a></li> <li><a href="/wiki/Funkcja_Carmichaela" title="Funkcja Carmichaela">funkcja lambda Carmichaela</a></li> <li><a href="/wiki/Funkcja_pierwsza_omega" title="Funkcja pierwsza omega"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> (funkcja pierwsza omega)</a></li> <li><a href="/wiki/Funkcja_pierwsza_omega" title="Funkcja pierwsza omega"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span> (funkcja druga omega)</a></li> <li><a href="/wiki/Funkcja_Kempnera" title="Funkcja Kempnera">funkcja Kempnera</a></li> <li><a href="/wiki/Podsilnia" title="Podsilnia">podsilnia</a></li> <li><a href="/wiki/Suma_alikwotowa" title="Suma alikwotowa">suma alikwotowa</a></li> <li><a href="/wiki/Szereg_Grandiego" title="Szereg Grandiego">szereg Grandiego</a></li></ul> </td></tr></tbody></table></td></tr><tr class="a4"><th class="navbox-group opis" scope="row">inne przykłady<br />ciągów liczb</th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Charakter_Dirichleta" title="Charakter Dirichleta">charakter Dirichleta</a></li> <li><a href="/wiki/Ci%C4%85g_Fareya" title="Ciąg Fareya">ciąg Fareya</a></li> <li><a href="/wiki/Funkcja_von_Mangoldta" title="Funkcja von Mangoldta">funkcja von Mangoldta</a></li> <li><a href="/wiki/Liczby_Bernoulliego" title="Liczby Bernoulliego">liczby Bernoulliego</a></li> <li><a href="/wiki/Problem_bazylejski" title="Problem bazylejski">problem bazylejski</a></li> <li><a href="/wiki/Ci%C4%85g_harmoniczny" title="Ciąg harmoniczny">ciąg harmoniczny</a> <ul><li><a href="/wiki/U%C5%82amek_egipski" title="Ułamek egipski">ułamki egipskie</a></li></ul></li> <li><a href="/wiki/Wz%C3%B3r_Stirlinga" title="Wzór Stirlinga">wzór Stirlinga</a></li></ul> </td></tr><tr class="a5"><th class="navbox-group opis" scope="row"><a href="/wiki/Twierdzenie" title="Twierdzenie">twierdzenia</a></th><td class="navbox-list spis"><table class="inner-standard"><tbody><tr class="a5_1"><th class="navbox-group opis" scope="row">o <a href="/wiki/Granica_ci%C4%85gu" title="Granica ciągu">granicach</a></th><td class="navbox-list spis hlist navbox-even"> <ul><li><a class="mw-selflink selflink">Bolzana-Weierstrassa</a></li> <li><a href="/wiki/Twierdzenie_o_dw%C3%B3ch_ci%C4%85gach" title="Twierdzenie o dwóch ciągach">o dwóch ciągach</a></li> <li><a href="/wiki/Twierdzenie_o_trzech_ci%C4%85gach" title="Twierdzenie o trzech ciągach">o trzech ciągach</a></li> <li><a href="/wiki/Twierdzenie_o_zbie%C5%BCno%C5%9Bci_ci%C4%85gu_monotonicznego" title="Twierdzenie o zbieżności ciągu monotonicznego">o zbieżności ciągu monotonicznego</a></li> <li><a href="/wiki/Twierdzenie_o_zbie%C5%BCno%C5%9Bci_%C5%9Brednich" title="Twierdzenie o zbieżności średnich">o zbieżności średnich</a></li> <li><a href="/wiki/Twierdzenie_Stolza" title="Twierdzenie Stolza">Stolza</a></li></ul> </td></tr><tr class="a5_2"><th class="navbox-group opis" scope="row">inne</th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Twierdzenie_o_ci%C4%85gach_jednomonotonicznych" title="Twierdzenie o ciągach jednomonotonicznych">o ciągach jednomonotonicznych</a></li> <li><a href="/wiki/To%C5%BCsamo%C5%9B%C4%87_Czebyszewa" title="Tożsamość Czebyszewa">nierówność Czebyszewa</a></li></ul> </td></tr></tbody></table></td></tr><tr class="a6"><th class="navbox-group opis" scope="row">powiązane pojęcia</th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Podci%C4%85g_(matematyka)" title="Podciąg (matematyka)">podciąg</a></li> <li><a href="/wiki/Ci%C4%85g_uog%C3%B3lniony" title="Ciąg uogólniony">ciąg uogólniony</a></li> <li><a href="/wiki/Splot_Dirichleta" title="Splot Dirichleta">splot Dirichleta</a></li> <li><a href="/wiki/Symbol_nieoznaczony" title="Symbol nieoznaczony">symbole nieoznaczone</a></li></ul> </td></tr></tbody></table></div></div> <style data-mw-deduplicate="TemplateStyles:r74016753">.mw-parser-output #normdaten>div+div{margin-top:0.5em}.mw-parser-output #normdaten>div>div{background:var(--background-color-neutral,#eaecf0);padding:.2em .5em}.mw-parser-output #normdaten ul{margin:0;padding:0}.mw-parser-output #normdaten ul li:first-child{padding-left:.5em;border-left:1px solid var(--border-color-base,#a2a9b1)}</style> <div id="normdaten" class="catlinks"><div class="normdaten-typ-fehlt"><div><a href="/wiki/Encyklopedia_internetowa" title="Encyklopedia internetowa">Encyklopedie internetowe</a> (<span class="description"><a href="/wiki/Twierdzenie" title="Twierdzenie">twierdzenie</a></span>):</div> <ul><li><a href="/wiki/Encyklopedia_Britannica" title="Encyklopedia Britannica">Britannica</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://www.britannica.com/topic/Bolzano-Weierstrass-property">topic/Bolzano-Weierstrass-property</a></span></li> <li><a href="https://www.wikidata.org/wiki/Property:P1296" class="extiw" title="d:Property:P1296">Catalana</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://www.enciclopedia.cat/ec-gec-0010999.xml">0010999</a></span>, <span class="uid"><a rel="nofollow" class="external text" href="https://www.enciclopedia.cat/ec-gec-0010998.xml">0010998</a></span></li></ul> </div></div></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Źródło: „<a dir="ltr" href="https://pl.wikipedia.org/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&oldid=74231318">https://pl.wikipedia.org/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&oldid=74231318</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Specjalna:Kategorie" title="Specjalna:Kategorie">Kategoria</a>: <ul><li><a href="/wiki/Kategoria:Twierdzenia_o_ci%C4%85gach_rzeczywistych" title="Kategoria:Twierdzenia o ciągach rzeczywistych">Twierdzenia o ciągach rzeczywistych</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Ukryta kategoria: <ul><li><a href="/wiki/Kategoria:Szablon_cytowania_ksi%C4%85%C5%BCki_%E2%80%93_brak_numeru_strony" title="Kategoria:Szablon cytowania książki – brak numeru strony">Szablon cytowania książki – brak numeru strony</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Tę stronę ostatnio edytowano 9 lip 2024, 12:45.</li> <li id="footer-info-copyright">Tekst udostępniany na licencji <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.pl">Creative Commons: uznanie autorstwa, na tych samych warunkach</a>, z możliwością obowiązywania dodatkowych ograniczeń. Zobacz szczegółowe informacje o <a class="external text" href="https://foundation.wikimedia.org/wiki/Policy:Terms_of_Use/pl">warunkach korzystania</a>.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Polityka prywatności</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:O_Wikipedii">O Wikipedii</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:Korzystasz_z_Wikipedii_tylko_na_w%C5%82asn%C4%85_odpowiedzialno%C5%9B%C4%87">Korzystasz z Wikipedii tylko na własną odpowiedzialność</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Powszechne Zasady Postępowania</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Dla deweloperów</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/pl.wikipedia.org">Statystyki</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Oświadczenie o ciasteczkach</a></li> <li id="footer-places-mobileview"><a href="//pl.m.wikipedia.org/w/index.php?title=Twierdzenie_Bolzana-Weierstrassa&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Wersja mobilna</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-694cf4987f-76n9j","wgBackendResponseTime":165,"wgPageParseReport":{"limitreport":{"cputime":"0.351","walltime":"0.564","ppvisitednodes":{"value":2458,"limit":1000000},"postexpandincludesize":{"value":37370,"limit":2097152},"templateargumentsize":{"value":998,"limit":2097152},"expansiondepth":{"value":9,"limit":100},"expensivefunctioncount":{"value":0,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":14306,"limit":5000000},"entityaccesscount":{"value":3,"limit":400},"timingprofile":["100.00% 329.225 1 -total"," 41.87% 137.838 5 Szablon:Szablon_nawigacyjny"," 41.24% 135.779 1 Szablon:Ciągi_liczbowe"," 33.94% 111.748 1 Szablon:Kontrola_autorytatywna"," 10.02% 32.978 3 Szablon:Cytuj_książkę"," 8.76% 28.838 1 Szablon:MathWorld"," 4.80% 15.796 1 Szablon:Uwagi"," 1.05% 3.454 1 Szablon:Otwarty_dostęp"," 0.71% 2.353 1 Szablon:Odn/id"]},"scribunto":{"limitreport-timeusage":{"value":"0.189","limit":"10.000"},"limitreport-memusage":{"value":2796538,"limit":52428800},"limitreport-logs":"required = table#1 {\n}\nrequired = table#1 {\n}\nrequired = table#1 {\n}\nrequired = table#1 {\n}\n\n== Szablon:Szablon nawigacyjny ==\n\n\n== Szablon:Szablon nawigacyjny ==\n\n\n== Szablon:Szablon nawigacyjny ==\n\n\n== Szablon:Szablon nawigacyjny ==\n\n\n== Szablon:Szablon nawigacyjny ==\n\nrequired = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.eqiad.main-8f644c877-rj9zd","timestamp":"20241106051524","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Twierdzenie Bolzana-Weierstrassa","url":"https:\/\/pl.wikipedia.org\/wiki\/Twierdzenie_Bolzana-Weierstrassa","sameAs":"http:\/\/www.wikidata.org\/entity\/Q468391","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q468391","author":{"@type":"Organization","name":"Wsp\u00f3\u0142tw\u00f3rcy projekt\u00f3w Fundacji Wikimedia"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2006-05-01T19:20:24Z","headline":"twierdzenie o ci\u0105gach ograniczonych"}</script> </body> </html>

CINXE.COM

Twierdzenie Bolzana-Weierstrassa – Wikipedia, wolna encyklopedia