CINXE.COM

<!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="zh" dir="ltr"> <head> <meta charset="UTF-8"> <title>上界和下界 - 维基百科，自由的百科全书</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(/(?:^|; )zhwikimwclientpreferences=([^;]+)/);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":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"zh", "wgMonthNames":["","1月","2月","3月","4月","5月","6月","7月","8月","9月","10月","11月","12月"],"wgRequestId":"efc036bb-5313-4a91-847d-b5ee3187ad16","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"上界和下界","wgTitle":"上界和下界","wgCurRevisionId":72859819,"wgRevisionId":72859819,"wgArticleId":35361,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["全部小作品","逻辑学小作品","序理论"],"wgPageViewLanguage":"zh","wgPageContentLanguage":"zh","wgPageContentModel":"wikitext","wgRelevantPageName":"上界和下界","wgRelevantArticleId":35361,"wgUserVariant":"zh","wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":true,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{ "pageLanguageCode":"zh","pageLanguageDir":"ltr","pageVariantFallbacks":["zh-hans","zh-hant","zh-cn","zh-tw","zh-hk","zh-sg","zh-mo","zh-my"]},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":1000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q13222579","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.gadget.large-font":"ready","ext.globalCssJs.user.styles":"ready","site.styles":"ready", "user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","ext.cite.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"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","site","mediawiki.page.ready","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.edit0","ext.gadget.WikiMiniAtlas","ext.gadget.UnihanTooltips","ext.gadget.Difflink","ext.gadget.pseudonamespace-UI","ext.gadget.SpecialWikitext","ext.gadget.switcher","ext.gadget.VariantAlly","ext.gadget.AdvancedSiteNotices","ext.gadget.hideConversionTab","ext.gadget.internalLinkHelper-altcolor","ext.gadget.noteTA","ext.gadget.NavFrame","ext.gadget.collapsibleTables","ext.gadget.scrollUpButton", "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=zh&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.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=zh&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=zh&modules=ext.gadget.large-font&only=styles&skin=vector-2022"> <link rel="stylesheet" href="/w/load.php?lang=zh&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="上界和下界 - 维基百科，自由的百科全书"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//zh.m.wikipedia.org/wiki/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C"> <link rel="alternate" type="application/x-wiki" title="编辑本页" href="/w/index.php?title=%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C&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 (zh)"> <link rel="EditURI" type="application/rsd+xml" href="//zh.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://zh.wikipedia.org/wiki/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C"> <link rel="alternate" hreflang="zh" href="https://zh.wikipedia.org/wiki/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C"> <link rel="alternate" hreflang="zh-Hans" href="https://zh.wikipedia.org/zh-hans/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C"> <link rel="alternate" hreflang="zh-Hans-CN" href="https://zh.wikipedia.org/zh-cn/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C"> <link rel="alternate" hreflang="zh-Hans-MY" href="https://zh.wikipedia.org/zh-my/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C"> <link rel="alternate" hreflang="zh-Hans-SG" href="https://zh.wikipedia.org/zh-sg/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C"> <link rel="alternate" hreflang="zh-Hant" href="https://zh.wikipedia.org/zh-hant/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C"> <link rel="alternate" hreflang="zh-Hant-HK" href="https://zh.wikipedia.org/zh-hk/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C"> <link rel="alternate" hreflang="zh-Hant-MO" href="https://zh.wikipedia.org/zh-mo/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C"> <link rel="alternate" hreflang="zh-Hant-TW" href="https://zh.wikipedia.org/zh-tw/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C"> <link rel="alternate" hreflang="x-default" href="https://zh.wikipedia.org/wiki/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.zh"> <link rel="alternate" type="application/atom+xml" title="Wikipedia的Atom feed" href="/w/index.php?title=Special:%E6%9C%80%E8%BF%91%E6%9B%B4%E6%94%B9&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-上界和下界 rootpage-上界和下界 skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">跳转到内容</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="站点"> <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="主菜单" > <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">主菜单</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">主菜单</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">隐藏</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> 导航 </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:%E9%A6%96%E9%A1%B5" title="访问首页[z]" accesskey="z"><span>首页</span></a></li><li id="n-indexpage" class="mw-list-item"><a href="/wiki/Wikipedia:%E5%88%86%E7%B1%BB%E7%B4%A2%E5%BC%95" title="以分类索引搜寻中文维基百科"><span>分类索引</span></a></li><li id="n-Featured_content" class="mw-list-item"><a href="/wiki/Portal:%E7%89%B9%E8%89%B2%E5%85%A7%E5%AE%B9" title="查看中文维基百科的特色内容"><span>特色内容</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:%E6%96%B0%E8%81%9E%E5%8B%95%E6%85%8B" title="提供当前新闻事件的背景资料"><span>新闻动态</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:%E6%9C%80%E8%BF%91%E6%9B%B4%E6%94%B9" title="列出维基百科中的最近修改[r]" accesskey="r"><span>最近更改</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:%E9%9A%8F%E6%9C%BA%E9%A1%B5%E9%9D%A2" title="随机载入一个页面[x]" accesskey="x"><span>随机条目</span></a></li> </ul> </div> </div> <div id="p-help" class="vector-menu mw-portlet mw-portlet-help" > <div class="vector-menu-heading"> 帮助 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:%E7%9B%AE%E5%BD%95" title="寻求帮助"><span>帮助</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:%E7%A4%BE%E7%BE%A4%E9%A6%96%E9%A1%B5" title="关于本计划、你可以做什么、应该如何做"><span>维基社群</span></a></li><li id="n-policy" class="mw-list-item"><a href="/wiki/Wikipedia:%E6%96%B9%E9%87%9D%E8%88%87%E6%8C%87%E5%BC%95" title="查看维基百科的方针和指引"><span>方针与指引</span></a></li><li id="n-villagepump" class="mw-list-item"><a href="/wiki/Wikipedia:%E4%BA%92%E5%8A%A9%E5%AE%A2%E6%A0%88" title="参与维基百科社群的讨论"><span>互助客栈</span></a></li><li id="n-Information_desk" class="mw-list-item"><a href="/wiki/Wikipedia:%E7%9F%A5%E8%AF%86%E9%97%AE%E7%AD%94" title="解答任何与维基百科无关的问题的地方"><span>知识问答</span></a></li><li id="n-conversion" class="mw-list-item"><a href="/wiki/Wikipedia:%E5%AD%97%E8%AF%8D%E8%BD%AC%E6%8D%A2" title="提出字词转换请求"><span>字词转换</span></a></li><li id="n-IRC" class="mw-list-item"><a href="/wiki/Wikipedia:IRC%E8%81%8A%E5%A4%A9%E9%A2%91%E9%81%93"><span>IRC即时聊天</span></a></li><li id="n-contact" class="mw-list-item"><a href="/wiki/Wikipedia:%E8%81%94%E7%BB%9C%E6%88%91%E4%BB%AC" title="如何联络维基百科"><span>联络我们</span></a></li><li id="n-about" class="mw-list-item"><a href="/wiki/Wikipedia:%E5%85%B3%E4%BA%8E" title="查看维基百科的简介"><span>关于维基百科</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Wikipedia:%E9%A6%96%E9%A1%B5" 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="维基百科" src="/static/images/mobile/copyright/wikipedia-wordmark-zh.svg" style="width: 6.5625em; height: 1.375em;"> <img class="mw-logo-tagline" alt="自由的百科全书" src="/static/images/mobile/copyright/wikipedia-tagline-zh.svg" width="103" height="14" style="width: 6.4375em; height: 0.875em;"> </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/Special:%E6%90%9C%E7%B4%A2" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="搜索维基百科[f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>搜索</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="搜索维基百科" aria-label="搜索维基百科" autocapitalize="sentences" title="搜索维基百科[f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:搜索"> </div> <button class="cdx-button cdx-search-input__end-button">搜索</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="个人工具"> <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="外观"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="更改页面字体大小、宽度和颜色的外观" > <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="外观" > <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">外观</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="https://donate.wikimedia.org/?utm_source=donate&utm_medium=sidebar&utm_campaign=spontaneous&uselang=zh-hans" class=""><span>资助维基百科</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=Special:%E5%88%9B%E5%BB%BA%E8%B4%A6%E6%88%B7&returnto=%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C" title="我们推荐您创建账号并登录，但这不是强制性的" class=""><span>创建账号</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=Special:%E7%94%A8%E6%88%B7%E7%99%BB%E5%BD%95&returnto=%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C" title="建议你登录，尽管并非必须。[o]" accesskey="o" class=""><span>登录</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="更多选项" > <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="个人工具" > <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">个人工具</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="用户菜单" > <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="https://donate.wikimedia.org/?utm_source=donate&utm_medium=sidebar&utm_campaign=spontaneous&uselang=zh-hans"><span>资助维基百科</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:%E5%88%9B%E5%BB%BA%E8%B4%A6%E6%88%B7&returnto=%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C" title="我们推荐您创建账号并登录，但这不是强制性的"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>创建账号</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:%E7%94%A8%E6%88%B7%E7%99%BB%E5%BD%95&returnto=%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C" title="建议你登录，尽管并非必须。[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>登录</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"> 未登录编辑者的页面 <a href="/wiki/Help:%E6%96%B0%E6%89%8B%E5%85%A5%E9%97%A8" aria-label="了解有关编辑的更多信息"><span>了解详情</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/Special:%E6%88%91%E7%9A%84%E8%B4%A1%E7%8C%AE" title="来自此IP地址的编辑列表[y]" accesskey="y"><span>贡献</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%AE%A8%E8%AE%BA%E9%A1%B5" title="对于来自此IP地址编辑的讨论[n]" accesskey="n"><span>讨论</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="站点"> <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="目录" 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">目录</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">隐藏</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">序言</div> </a> </li> <li id="toc-性質" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#性質"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>性質</span> </div> </a> <ul id="toc-性質-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参见" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参见"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>参见</span> </div> </a> <ul id="toc-参见-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="目录" 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="开关目录" > <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">开关目录</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">上界和下界</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="前往另一种语言写成的文章。17种语言可用" > <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-17" 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">17种语言</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/%D8%AD%D8%AF%D9%88%D8%AF_%D8%B9%D9%84%D9%8A%D8%A7_%D9%88%D8%AF%D9%86%D9%8A%D8%A7" title="حدود عليا ودنيا – 阿拉伯语" lang="ar" hreflang="ar" data-title="حدود عليا ودنيا" data-language-autonym="العربية" data-language-local-name="阿拉伯语" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Fita_superior" title="Fita superior – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Fita superior" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Upper_and_lower_bounds" title="Upper and lower bounds – 英语" lang="en" hreflang="en" data-title="Upper and lower bounds" data-language-autonym="English" data-language-local-name="英语" 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/Elemento_mayorante_y_minorante" title="Elemento mayorante y minorante – 西班牙语" lang="es" hreflang="es" data-title="Elemento mayorante y minorante" data-language-autonym="Español" data-language-local-name="西班牙语" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Goi-borne" title="Goi-borne – 巴斯克语" lang="eu" hreflang="eu" data-title="Goi-borne" data-language-autonym="Euskara" data-language-local-name="巴斯克语" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Majorant_ou_minorant" title="Majorant ou minorant – 法语" lang="fr" hreflang="fr" data-title="Majorant ou minorant" data-language-autonym="Français" data-language-local-name="法语" 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/Elemento_maiorante_e_minorante" title="Elemento maiorante e minorante – 加利西亚语" lang="gl" hreflang="gl" data-title="Elemento maiorante e minorante" data-language-autonym="Galego" data-language-local-name="加利西亚语" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%A1%D7%9D_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="חסם (מתמטיקה) – 希伯来语" lang="he" hreflang="he" data-title="חסם (מתמטיקה)" data-language-autonym="עברית" data-language-local-name="希伯来语" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Gornja_i_donja_me%C4%91a" title="Gornja i donja međa – 克罗地亚语" lang="hr" hreflang="hr" data-title="Gornja i donja međa" data-language-autonym="Hrvatski" data-language-local-name="克罗地亚语" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Maggiorante_e_minorante" title="Maggiorante e minorante – 意大利语" lang="it" hreflang="it" data-title="Maggiorante e minorante" data-language-autonym="Italiano" data-language-local-name="意大利语" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja badge-Q70894304 mw-list-item" title=""><a href="https://ja.wikipedia.org/wiki/%E4%B8%8A%E7%95%8C%E3%81%A8%E4%B8%8B%E7%95%8C" title="上界と下界 – 日语" lang="ja" hreflang="ja" data-title="上界と下界" data-language-autonym="日本語" data-language-local-name="日语" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Maggiorant_e_minorant" title="Maggiorant e minorant – 倫巴底文" lang="lmo" hreflang="lmo" data-title="Maggiorant e minorant" data-language-autonym="Lombard" data-language-local-name="倫巴底文" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Bovengrens_en_ondergrens" title="Bovengrens en ondergrens – 荷兰语" lang="nl" hreflang="nl" data-title="Bovengrens en ondergrens" data-language-autonym="Nederlands" data-language-local-name="荷兰语" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Skranke_(matematikk)" title="Skranke (matematikk) – 书面挪威语" lang="nb" hreflang="nb" data-title="Skranke (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="书面挪威语" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%9Cst_ve_alt_s%C4%B1n%C4%B1r" title="Üst ve alt sınır – 土耳其语" lang="tr" hreflang="tr" data-title="Üst ve alt sınır" data-language-autonym="Türkçe" data-language-local-name="土耳其语" 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%92%D0%B5%D1%80%D1%85%D0%BD%D1%8F_%D1%82%D0%B0_%D0%BD%D0%B8%D0%B6%D0%BD%D1%8F_%D0%BC%D0%B5%D0%B6%D0%B0" title="Верхня та нижня межа – 乌克兰语" lang="uk" hreflang="uk" data-title="Верхня та нижня межа" data-language-autonym="Українська" data-language-local-name="乌克兰语" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%95%8C%E9%99%90_(%E6%95%B8%E5%AD%B8)" title="界限 (數學) – 粤语" lang="yue" hreflang="yue" data-title="界限 (數學)" data-language-autonym="粵語" data-language-local-name="粤语" class="interlanguage-link-target"><span>粵語</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/Q13222579#sitelinks-wikipedia" title="编辑跨语言链接" class="wbc-editpage">编辑链接</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="命名空间"> <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/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C" title="浏览条目正文[c]" accesskey="c"><span>条目</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C" rel="discussion" title="关于此页面的讨论[t]" accesskey="t"><span>讨论</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown " > <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="更改语言变体" > <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">不转换</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-varlang-0" class="selected ca-variants-zh mw-list-item"><a href="/zh/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C" lang="zh" hreflang="zh"><span>不转换</span></a></li><li id="ca-varlang-1" class="ca-variants-zh-Hans mw-list-item"><a href="/zh-hans/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C" lang="zh-Hans" hreflang="zh-Hans"><span>简体</span></a></li><li id="ca-varlang-2" class="ca-variants-zh-Hant mw-list-item"><a href="/zh-hant/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C" lang="zh-Hant" hreflang="zh-Hant"><span>繁體</span></a></li><li id="ca-varlang-3" class="ca-variants-zh-Hans-CN mw-list-item"><a href="/zh-cn/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C" lang="zh-Hans-CN" hreflang="zh-Hans-CN"><span>大陆简体</span></a></li><li id="ca-varlang-4" class="ca-variants-zh-Hant-HK mw-list-item"><a href="/zh-hk/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C" lang="zh-Hant-HK" hreflang="zh-Hant-HK"><span>香港繁體</span></a></li><li id="ca-varlang-5" class="ca-variants-zh-Hant-MO mw-list-item"><a href="/zh-mo/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C" lang="zh-Hant-MO" hreflang="zh-Hant-MO"><span>澳門繁體</span></a></li><li id="ca-varlang-6" class="ca-variants-zh-Hans-MY mw-list-item"><a href="/zh-my/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C" lang="zh-Hans-MY" hreflang="zh-Hans-MY"><span>大马简体</span></a></li><li id="ca-varlang-7" class="ca-variants-zh-Hans-SG mw-list-item"><a href="/zh-sg/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C" lang="zh-Hans-SG" hreflang="zh-Hans-SG"><span>新加坡简体</span></a></li><li id="ca-varlang-8" class="ca-variants-zh-Hant-TW mw-list-item"><a href="/zh-tw/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C" lang="zh-Hant-TW" hreflang="zh-Hant-TW"><span>臺灣正體</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="查看"> <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/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C"><span>阅读</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C&action=edit" title="编辑该页面[e]" accesskey="e"><span>编辑</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C&action=history" title="本页面的早前版本。[h]" accesskey="h"><span>查看历史</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="页面工具"> <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="工具" > <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">工具</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">工具</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">隐藏</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="更多选项" > <div class="vector-menu-heading"> 操作 </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/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C"><span>阅读</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C&action=edit" title="编辑该页面[e]" accesskey="e"><span>编辑</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C&action=history"><span>查看历史</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> 常规 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:%E9%93%BE%E5%85%A5%E9%A1%B5%E9%9D%A2/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C" title="列出所有与本页相链的页面[j]" accesskey="j"><span>链入页面</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:%E9%93%BE%E5%87%BA%E6%9B%B4%E6%94%B9/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C" rel="nofollow" title="页面链出所有页面的更改[k]" accesskey="k"><span>相关更改</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Project:%E4%B8%8A%E4%BC%A0" title="上传图像或多媒体文件[u]" accesskey="u"><span>上传文件</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:%E7%89%B9%E6%AE%8A%E9%A1%B5%E9%9D%A2" title="全部特殊页面的列表[q]" accesskey="q"><span>特殊页面</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C&oldid=72859819" title="此页面该修订版本的固定链接"><span>固定链接</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C&action=info" title="关于此页面的更多信息"><span>页面信息</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:%E5%BC%95%E7%94%A8%E6%AD%A4%E9%A1%B5%E9%9D%A2&page=%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C&id=72859819&wpFormIdentifier=titleform" title="有关如何引用此页面的信息"><span>引用此页</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:URL%E7%BC%A9%E7%9F%AD%E7%A8%8B%E5%BA%8F&url=https%3A%2F%2Fzh.wikipedia.org%2Fwiki%2F%25E4%25B8%258A%25E7%2595%258C%25E5%2592%258C%25E4%25B8%258B%25E7%2595%258C"><span>获取短链接</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fzh.wikipedia.org%2Fwiki%2F%25E4%25B8%258A%25E7%2595%258C%25E5%2592%258C%25E4%25B8%258B%25E7%2595%258C"><span>下载二维码</span></a></li> </ul> </div> </div> <div id="p-electronpdfservice-sidebar-portlet-heading" class="vector-menu mw-portlet mw-portlet-electronpdfservice-sidebar-portlet-heading" > <div class="vector-menu-heading"> 打印/导出 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="electron-print_pdf" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C&action=show-download-screen"><span>下载为PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="javascript:print();" rel="alternate" title="本页面的可打印版本[p]" accesskey="p"><span>打印页面</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"> 在其他项目中 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q13222579" title="链接到连接的数据仓库项目[g]" accesskey="g"><span>维基数据项目</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="页面工具"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="外观"> <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">外观</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">隐藏</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">维基百科，自由的百科全书</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="zh" dir="ltr"><p>設<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,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dccbed0945476c5f14b21d76e3d936f4acf547b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.394ex; height:2.843ex;" alt="{\displaystyle (A,\leq )}"></span>為一個<a href="/wiki/%E5%81%8F%E5%BA%8F%E9%9B%86" class="mw-redirect" title="偏序集">偏序集</a>，若存在<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 y\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7967ddfa570a9b0cc13b4785f57407d2526cf292" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.739ex; height:2.509ex;" alt="{\displaystyle y\in A}"></span>，能滿足<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 \forall x\in B\subseteq A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mi>B</mi> <mo>⊆</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\in B\subseteq A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/186201065c6bc5f4bf2b60a4abfd2ddfcd614905" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.068ex; height:2.343ex;" alt="{\displaystyle \forall x\in B\subseteq A}"></span>都有<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 x\leq y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07a0bc023490be1c08e6c33a9cdc93bec908224" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.343ex;" alt="{\displaystyle x\leq y}"></span>，則<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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>稱作集合<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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>的<b>上界</b>，若存在<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 z\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40028fe140831e54037c7b9f8d6907fdeaf4c256" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.672ex; height:2.176ex;" alt="{\displaystyle z\in A}"></span>，能滿足<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 \forall x\in B\subseteq A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mi>B</mi> <mo>⊆</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\in B\subseteq A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/186201065c6bc5f4bf2b60a4abfd2ddfcd614905" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.068ex; height:2.343ex;" alt="{\displaystyle \forall x\in B\subseteq A}"></span>都有<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 x\geq z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≥</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\geq z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1731ac4ae3d339367ac478682728e3253e2148bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.516ex; height:2.176ex;" alt="{\displaystyle x\geq z}"></span>，則<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 z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>稱作<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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>的<b>下界</b>。 </p><p>例如在<a href="/wiki/%E5%AE%9E%E5%8F%98%E5%87%BD%E6%95%B0%E8%AE%BA" title="实变函数论">實變數</a>中，若存在一個<a href="/wiki/%E5%AF%A6%E6%95%B8" class="mw-redirect" title="實數">實數</a><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>，能滿足<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 \forall x\in S\subseteq R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mi>S</mi> <mo>⊆</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\in S\subseteq R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68964c8d087600721de6199522b58a0737d99b7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.825ex; height:2.343ex;" alt="{\displaystyle \forall x\in S\subseteq R}"></span>都有<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 x\leq b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d1803bf9d4c215835aab217a666aa8225f1626a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.426ex; height:2.343ex;" alt="{\displaystyle x\leq b}"></span>，則<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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>即為集合<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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>的<b>上界</b>，若存在一個<a href="/wiki/%E5%AF%A6%E6%95%B8" class="mw-redirect" title="實數">實數</a><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span>，能滿足<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 \forall x\in S\subseteq R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mi>S</mi> <mo>⊆</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\in S\subseteq R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68964c8d087600721de6199522b58a0737d99b7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.825ex; height:2.343ex;" alt="{\displaystyle \forall x\in S\subseteq R}"></span>都有<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 x\geq c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≥</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\geq c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07f2e5b1cc52ebf6083048058c1be34c0939dbf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.435ex; height:2.176ex;" alt="{\displaystyle x\geq c}"></span>，則<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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span>即為集合<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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>的<b>下界</b>。 </p> <div class="mw-heading mw-heading2"><h2 id="性質"><span id=".E6.80.A7.E8.B3.AA"></span>性質</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C&action=edit&section=1" title="编辑章节：性質"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>连续性公理</b>：在非空<a href="/wiki/%E5%AE%9E%E6%95%B0%E9%9B%86" class="mw-redirect" title="实数集">实数集</a>中，若含上界，則必含<a href="/wiki/%E6%9C%80%E5%B0%8F%E4%B8%8A%E7%95%8C" title="最小上界">最小上界</a>（<b>上确界</b>）；若含下界，則必存在<a href="/wiki/%E6%9C%80%E5%A4%A7%E4%B8%8B%E7%95%8C" title="最大下界">最大下界</a>（<b>下确界</b>)。<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="参见"><span id=".E5.8F.82.E8.A7.81"></span>参见</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C&action=edit&section=2" title="编辑章节：参见"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E5%81%8F%E5%BA%8F%E5%85%B3%E7%B3%BB" title="偏序关系">偏序关系</a></li> <li><a href="/wiki/%E6%9C%80%E5%B0%8F%E4%B8%8A%E7%95%8C" title="最小上界">最小上界</a></li> <li><a href="/wiki/%E6%9C%80%E5%A4%A7%E4%B8%8B%E7%95%8C" title="最大下界">最大下界</a></li></ul> <table class="metadata plainlinks stub" role="presentation" style="background:transparent"><tbody><tr class="noresize"><td><span typeof="mw:File"><a href="/wiki/File:Formal_logic_template.svg" class="mw-file-description"><img alt="小作品圖示" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Formal_logic_template.svg/30px-Formal_logic_template.svg.png" decoding="async" width="30" height="30" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Formal_logic_template.svg/45px-Formal_logic_template.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/Formal_logic_template.svg/60px-Formal_logic_template.svg.png 2x" data-file-width="48" data-file-height="48" /></a></span></td><td style="font-size:smaller">这是一篇與<a href="/wiki/%E9%82%8F%E8%BC%AF" class="mw-redirect" title="邏輯">逻辑学</a>相關的<a href="/wiki/Wikipedia:%E5%B0%8F%E4%BD%9C%E5%93%81" title="Wikipedia:小作品">小作品</a>。您可以通过<a class="external text" href="https://zh.wikipedia.org/w/index.php?title=%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C&action=edit">编辑或修订</a>扩充其内容。<style data-mw-deduplicate="TemplateStyles:r84265675">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:" :"}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist-pipe dd::after,.mw-parser-output .hlist-pipe li::after{content:" | ";font-weight:normal}.mw-parser-output .hlist-hyphen dd::after,.mw-parser-output .hlist-hyphen li::after{content:" - ";font-weight:normal}.mw-parser-output .hlist-comma dd::after,.mw-parser-output .hlist-comma li::after{content:"、";font-weight:normal}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:"（"counter(listitem)"\a0 "}.mw-parser-output ul.cslist,.mw-parser-output ul.sslist{margin:0;padding:0;display:inline-block;list-style:none}.mw-parser-output .cslist li,.mw-parser-output .sslist li{margin:0;display:inline-block}.mw-parser-output .cslist li::after{content:"，"}.mw-parser-output .sslist li::after{content:"；"}.mw-parser-output .cslist li:last-child::after,.mw-parser-output .sslist li:last-child::after{content:none}</style><style data-mw-deduplicate="TemplateStyles:r84244141">.mw-parser-output .navbar{display:inline;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:110%;margin:0 8em}.mw-parser-output .navbar-ct-mini{font-size:110%;margin:0 5em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini" style="position: absolute; right: 15px; display: none;"><ul><li class="nv-view"><a href="/wiki/Template:Logic-stub" title="Template:Logic-stub"><abbr title="查看该模板">查</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Logic-stub" title="Template talk:Logic-stub"><abbr title="讨论该模板">论</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:%E7%BC%96%E8%BE%91%E9%A1%B5%E9%9D%A2/Template:Logic-stub" title="Special:编辑页面/Template:Logic-stub"><abbr title="编辑该模板">编</abbr></a></li></ul></div></td></tr></tbody></table> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="http://wiki.cnki.com.cn/HotWord/4734974.htm">确界存在定理-学术百科-知网空间</a>. wiki.cnki.com.cn. 知网空间. <span class="reference-accessdate"> [<span class="nowrap">2017-06-08</span>]</span>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20201028002154/http://wiki.cnki.com.cn/HotWord/4734974.htm">存档</a>于2020-10-28）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C&rft.atitle=%E7%A1%AE%E7%95%8C%E5%AD%98%E5%9C%A8%E5%AE%9A%E7%90%86-%E5%AD%A6%E6%9C%AF%E7%99%BE%E7%A7%91-%E7%9F%A5%E7%BD%91%E7%A9%BA%E9%97%B4&rft.genre=unknown&rft.jtitle=wiki.cnki.com.cn&rft_id=http%3A%2F%2Fwiki.cnki.com.cn%2FHotWord%2F4734974.htm&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol>    </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="">检索自“<a dir="ltr" href="https://zh.wikipedia.org/w/index.php?title=上界和下界&oldid=72859819">https://zh.wikipedia.org/w/index.php?title=上界和下界&oldid=72859819</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Special:%E9%A1%B5%E9%9D%A2%E5%88%86%E7%B1%BB" title="Special:页面分类">分类</a>：<ul><li><a href="/wiki/Category:%E5%BA%8F%E7%90%86%E8%AE%BA" title="Category:序理论">序理论</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">隐藏分类：<ul><li><a href="/wiki/Category:%E5%85%A8%E9%83%A8%E5%B0%8F%E4%BD%9C%E5%93%81" title="Category:全部小作品">全部小作品</a></li><li><a href="/wiki/Category:%E9%80%BB%E8%BE%91%E5%AD%A6%E5%B0%8F%E4%BD%9C%E5%93%81" title="Category:逻辑学小作品">逻辑学小作品</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"> 本页面最后修订于2022年7月23日 (星期六) 23:47。</li> <li id="footer-info-copyright">本站的全部文字在<a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/deed.zh">知识共享署名-相同方式共享 4.0协议</a>之条款下提供，附加条款亦可能应用。（请参阅<a class="external text" href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use">使用条款</a>）<br /> Wikipedia®和维基百科标志是<a rel="nofollow" class="external text" href="https://wikimediafoundation.org/zh">维基媒体基金会</a>的注册商标；维基™是维基媒体基金会的商标。<br /> 维基媒体基金会是按美国国內稅收法501(c)(3)登记的<a class="external text" href="https://donate.wikimedia.org/wiki/Special:MyLanguage/Tax_deductibility">非营利慈善机构</a>。<br /></li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">隐私政策</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:%E5%85%B3%E4%BA%8E">关于维基百科</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:%E5%85%8D%E8%B4%A3%E5%A3%B0%E6%98%8E">免责声明</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">行为准则</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">开发者</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/zh.wikipedia.org">统计</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie声明</a></li> <li id="footer-places-mobileview"><a href="//zh.m.wikipedia.org/w/index.php?title=%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">手机版视图</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-57488d5c7d-6v6z6","wgBackendResponseTime":168,"wgPageParseReport":{"limitreport":{"cputime":"0.139","walltime":"0.226","ppvisitednodes":{"value":192,"limit":1000000},"postexpandincludesize":{"value":5017,"limit":2097152},"templateargumentsize":{"value":0,"limit":2097152},"expansiondepth":{"value":6,"limit":100},"expensivefunctioncount":{"value":0,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":4338,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 128.857 1 -total"," 60.64% 78.142 1 Template:Logic-stub"," 57.97% 74.704 1 Template:Asbox"," 33.16% 42.732 1 Template:Cite_web"]},"scribunto":{"limitreport-timeusage":{"value":"0.077","limit":"10.000"},"limitreport-memusage":{"value":1691077,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-77bc9d5b54-r2kxg","timestamp":"20241118110352","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"\u4e0a\u754c\u548c\u4e0b\u754c","url":"https:\/\/zh.wikipedia.org\/wiki\/%E4%B8%8A%E7%95%8C%E5%92%8C%E4%B8%8B%E7%95%8C","sameAs":"http:\/\/www.wikidata.org\/entity\/Q13222579","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q13222579","author":{"@type":"Organization","name":"\u7ef4\u57fa\u5a92\u4f53\u9879\u76ee\u8d21\u732e\u8005"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2004-07-01T10:41:33Z","dateModified":"2022-07-23T23:47:37Z"}</script> </body> </html>