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Disponibil în 38 limbi" > <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-38" 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">38 limbi</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Archimedisches_Axiom" title="Archimedisches Axiom – germană (Elveția)" lang="gsw" hreflang="gsw" data-title="Archimedisches Axiom" data-language-autonym="Alemannisch" data-language-local-name="germană (Elveția)" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AE%D8%A7%D8%B5%D9%8A%D8%A9_%D8%A3%D8%B1%D8%AE%D9%85%D9%8A%D8%AF%D8%B3" title="خاصية أرخميدس – arabă" lang="ar" hreflang="ar" data-title="خاصية أرخميدس" data-language-autonym="العربية" data-language-local-name="arabă" 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/Arximed_aksiomu" title="Arximed aksiomu – azeră" lang="az" hreflang="az" data-title="Arximed aksiomu" data-language-autonym="Azərbaycanca" data-language-local-name="azeră" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%BA%D1%81%D1%96%D1%91%D0%BC%D0%B0_%D0%90%D1%80%D1%85%D1%96%D0%BC%D0%B5%D0%B4%D0%B0" title="Аксіёма Архімеда – belarusă" lang="be" hreflang="be" data-title="Аксіёма Архімеда" data-language-autonym="Беларуская" data-language-local-name="belarusă" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Axioma_d%27Arquimedes" title="Axioma d'Arquimedes – catalană" lang="ca" hreflang="ca" data-title="Axioma d'Arquimedes" data-language-autonym="Català" data-language-local-name="catalană" 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/Archim%C3%A9d%C5%AFv_axiom" title="Archimédův axiom – cehă" lang="cs" hreflang="cs" data-title="Archimédův axiom" data-language-autonym="Čeština" data-language-local-name="cehă" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%90%D1%80%D1%85%D0%B8%D0%BC%D0%B5%D0%B4_%D0%B0%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D0%B8" title="Архимед аксиоми – ciuvașă" lang="cv" hreflang="cv" data-title="Архимед аксиоми" data-language-autonym="Чӑвашла" data-language-local-name="ciuvașă" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Archimedisches_Axiom" title="Archimedisches Axiom – germană" lang="de" hreflang="de" data-title="Archimedisches Axiom" data-language-autonym="Deutsch" data-language-local-name="germană" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CF%81%CF%87%CE%B9%CE%BC%CE%AE%CE%B4%CE%B5%CE%B9%CE%B1_%CE%B9%CE%B4%CE%B9%CF%8C%CF%84%CE%B7%CF%84%CE%B1" title="Αρχιμήδεια ιδιότητα – greacă" lang="el" hreflang="el" data-title="Αρχιμήδεια ιδιότητα" data-language-autonym="Ελληνικά" data-language-local-name="greacă" 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/Archimedean_property" title="Archimedean property – engleză" lang="en" hreflang="en" data-title="Archimedean property" data-language-autonym="English" data-language-local-name="engleză" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Ar%C4%A5imeda_eco" title="Arĥimeda eco – esperanto" lang="eo" hreflang="eo" data-title="Arĥimeda eco" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Axioma_de_Arqu%C3%ADmedes" title="Axioma de Arquímedes – spaniolă" lang="es" hreflang="es" data-title="Axioma de Arquímedes" data-language-autonym="Español" data-language-local-name="spaniolă" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Archimedese_aksioom" title="Archimedese aksioom – estonă" lang="et" hreflang="et" data-title="Archimedese aksioom" data-language-autonym="Eesti" data-language-local-name="estonă" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AE%D8%A7%D8%B5%DB%8C%D8%AA_%D8%A7%D8%B1%D8%B4%D9%85%DB%8C%D8%AF%D8%B3%DB%8C" title="خاصیت ارشمیدسی – persană" lang="fa" hreflang="fa" data-title="خاصیت ارشمیدسی" data-language-autonym="فارسی" data-language-local-name="persană" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Arkhimedeen_lause" title="Arkhimedeen lause – finlandeză" lang="fi" hreflang="fi" data-title="Arkhimedeen lause" data-language-autonym="Suomi" data-language-local-name="finlandeză" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Archim%C3%A9dien" title="Archimédien – franceză" lang="fr" hreflang="fr" data-title="Archimédien" data-language-autonym="Français" data-language-local-name="franceză" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%9B%D7%95%D7%A0%D7%AA_%D7%90%D7%A8%D7%9B%D7%99%D7%9E%D7%93%D7%A1" title="תכונת ארכימדס – ebraică" lang="he" hreflang="he" data-title="תכונת ארכימדס" data-language-autonym="עברית" data-language-local-name="ebraică" 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/Arhimedov_aksiom" title="Arhimedov aksiom – croată" lang="hr" hreflang="hr" data-title="Arhimedov aksiom" data-language-autonym="Hrvatski" data-language-local-name="croată" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D6%80%D6%84%D5%AB%D5%B4%D5%A5%D5%A4%D5%AB_%D5%A1%D6%84%D5%BD%D5%AB%D5%B8%D5%B4" title="Արքիմեդի աքսիոմ – armeană" lang="hy" hreflang="hy" data-title="Արքիմեդի աքսիոմ" data-language-autonym="Հայերեն" data-language-local-name="armeană" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Sifat_Archimedes" title="Sifat Archimedes – indoneziană" lang="id" hreflang="id" data-title="Sifat Archimedes" data-language-autonym="Bahasa Indonesia" data-language-local-name="indoneziană" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%A2%E3%83%AB%E3%82%AD%E3%83%A1%E3%83%87%E3%82%B9%E3%81%AE%E6%80%A7%E8%B3%AA" title="アルキメデスの性質 – japoneză" lang="ja" hreflang="ja" data-title="アルキメデスの性質" data-language-autonym="日本語" data-language-local-name="japoneză" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%90%E1%83%A0%E1%83%A5%E1%83%98%E1%83%9B%E1%83%94%E1%83%93%E1%83%94%E1%83%A1_%E1%83%90%E1%83%A5%E1%83%A1%E1%83%98%E1%83%9D%E1%83%9B%E1%83%90" title="არქიმედეს აქსიომა – georgiană" lang="ka" hreflang="ka" data-title="არქიმედეს აქსიომა" data-language-autonym="ქართული" data-language-local-name="georgiană" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D1%80%D1%85%D0%B8%D0%BC%D0%B5%D0%B4_%D0%B0%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D0%B0%D1%81%D1%8B" title="Архимед аксиомасы – kazahă" lang="kk" hreflang="kk" data-title="Архимед аксиомасы" data-language-autonym="Қазақша" data-language-local-name="kazahă" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%95%84%EB%A5%B4%ED%82%A4%EB%A9%94%EB%8D%B0%EC%8A%A4_%EC%84%B1%EC%A7%88" title="아르키메데스 성질 – coreeană" lang="ko" hreflang="ko" data-title="아르키메데스 성질" data-language-autonym="한국어" data-language-local-name="coreeană" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D1%80%D1%85%D0%B8%D0%BC%D0%B5%D0%B4_%D0%B0%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D0%B0%D1%81%D1%8B" title="Архимед аксиомасы – kârgâză" lang="ky" hreflang="ky" data-title="Архимед аксиомасы" data-language-autonym="Кыргызча" data-language-local-name="kârgâză" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Archimedische_eigenschap" title="Archimedische eigenschap – neerlandeză" lang="nl" hreflang="nl" data-title="Archimedische eigenschap" data-language-autonym="Nederlands" data-language-local-name="neerlandeză" 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/Arkimedes%E2%80%99_aksiom" title="Arkimedes’ aksiom – norvegiană bokmål" lang="nb" hreflang="nb" data-title="Arkimedes’ aksiom" data-language-autonym="Norsk bokmål" data-language-local-name="norvegiană bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Aksjomat_Archimedesa" title="Aksjomat Archimedesa – poloneză" lang="pl" hreflang="pl" data-title="Aksjomat Archimedesa" data-language-autonym="Polski" data-language-local-name="poloneză" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Propriedade_arquimediana" title="Propriedade arquimediana – portugheză" lang="pt" hreflang="pt" data-title="Propriedade arquimediana" data-language-autonym="Português" data-language-local-name="portugheză" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%90%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D0%B0_%D0%90%D1%80%D1%85%D0%B8%D0%BC%D0%B5%D0%B4%D0%B0" title="Аксиома Архимеда – rusă" lang="ru" hreflang="ru" data-title="Аксиома Архимеда" data-language-autonym="Русский" data-language-local-name="rusă" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Archimedova_axi%C3%B3ma" title="Archimedova axióma – slovacă" lang="sk" hreflang="sk" data-title="Archimedova axióma" data-language-autonym="Slovenčina" data-language-local-name="slovacă" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Arhimedov_aksiom" title="Arhimedov aksiom – slovenă" lang="sl" hreflang="sl" data-title="Arhimedov aksiom" data-language-autonym="Slovenščina" data-language-local-name="slovenă" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Arkimedes%27_axiom" title="Arkimedes' axiom – suedeză" lang="sv" hreflang="sv" data-title="Arkimedes' axiom" data-language-autonym="Svenska" data-language-local-name="suedeză" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a 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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">Aspect</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mută în bara laterală</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ascunde</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">De la Wikipedia, enciclopedia liberă</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="ro" dir="ltr"><p><i>A nu se confunda cu <a href="/wiki/Principiul_lui_Arhimede" class="mw-redirect" title="Principiul lui Arhimede">principiul lui Arhimede</a> din <a href="/wiki/Hidrostatic%C4%83" title="Hidrostatică">hidrostatică</a>!</i> </p><p><b>Axioma lui <a href="/wiki/Arhimede" title="Arhimede">Arhimede</a></b> reprezintă o proprietate specifică anumitor <a href="/wiki/Grup_(matematic%C4%83)" title="Grup (matematică)">grupuri</a> și <a href="/wiki/Corp_(matematic%C4%83)" title="Corp (matematică)">corpuri</a> din teoria structurilor <a href="/wiki/Algebr%C4%83" title="Algebră">algebrice</a>. </p><p>Alte denumiri: </p> <ul><li><i>Lema (proprietatea) lui Arhimede</i></li> <li><i>Axioma continuității</i></li> <li><i>Axioma (teorema) lui Eudoxus</i>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Istoric">Istoric</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axioma_lui_Arhimede&veaction=edit&section=1" title="Modifică secțiunea: Istoric" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Axioma_lui_Arhimede&action=edit&section=1" title="Edit section's source code: Istoric"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Atribuit lui <a href="/wiki/Arhimede" title="Arhimede">Arhimede</a> (sec. III î.Hr.), <a href="/wiki/Axiom%C4%83" title="Axiomă">axioma</a> se regăsește în scrierile lui <a href="/w/index.php?title=Eudoxus&action=edit&redlink=1" class="new" title="Eudoxus — pagină inexistentă">Eudoxus</a> (secolul al IV-lea î.Hr. - Boyer & Merzbach, 1991), iar termenul este introdus de matematicianul austriac <a href="/wiki/Otto_Stolz" title="Otto Stolz">Otto Stolz</a> în 1883. </p> <div class="mw-heading mw-heading2"><h2 id="Enunț"><span id="Enun.C8.9B"></span>Enunț</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axioma_lui_Arhimede&veaction=edit&section=2" title="Modifică secțiunea: Enunț" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Axioma_lui_Arhimede&action=edit&section=2" title="Edit section's source code: Enunț"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Teoremă</i> (principiul sau axioma lui Arhimede). Pentru orice <a href="/wiki/Num%C4%83r_real" title="Număr real">numere reale</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 x,y\in \mathbb {R} \!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in \mathbb {R} \!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2533e64087dfde1da09bda9ce9857854c87a6543" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.348ex; width:7.998ex; height:2.509ex;" alt="{\displaystyle x,y\in \mathbb {R} \!}"></span> cu <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>0\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>></mo> <mn>0</mn> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y>0\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d77c04336cda6f083d2b7a7534de6510274794c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.294ex; width:5.324ex; height:2.509ex;" alt="{\displaystyle y>0\!}"></span> există <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 \mathbb {N} \!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} \!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9498d37575298da80cda112c0359d70b4811192a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.341ex; width:5.867ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} \!}"></span> cu <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 ny.\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>n</mi> <mi>y</mi> <mo>.</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq ny.\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10342f88a529cd003ac7337206861312eb39e3e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.204ex; width:7.442ex; height:2.343ex;" alt="{\displaystyle x\leq ny.\!}"></span> </p><p>Pentru a demonstra proprietatea lui Arhimede, se utilizează următoarea <a href="/wiki/Teorem%C4%83" title="Teoremă">teoremă</a>: </p><p><i>Teoremă</i>. Pentru orice număr real <i>x</i> există un <a href="/wiki/Num%C4%83r_natural" title="Număr natural">număr natural</a> <i>m</i> astfel încât să avem: </p> <dl><dd><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 x\leq m\leq x+1.\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>m</mi> <mo>≤</mo> <mi>x</mi> <mo>+</mo> <mn>1.</mn> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq m\leq x+1.\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71f52b83642ba0790406d0f0f87313d9e5529e89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-right: -0.204ex; width:15.363ex; height:2.343ex;" alt="{\displaystyle x\leq m\leq x+1.\!}"></span></dd></dl></dd></dl> <p><i>Demonstrație</i>. Fie <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\in \mathbb {R} \!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} \!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b47ef7877b812c78cf410b3d1dd2a27b60f690fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.348ex; width:5.809ex; height:2.176ex;" alt="{\displaystyle x\in \mathbb {R} \!}"></span> fixat. Presupunem că <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>n\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>></mo> <mi>n</mi> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x>n\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f30e55f0f865dd9650b0ddf6db1cacb7ebc777e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.341ex; width:5.776ex; height:1.843ex;" alt="{\displaystyle x>n\!}"></span> pentru orice <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 \mathbb {N} .\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>.</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} .\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98496f497fe1efc57e5b81e5a0f8a0bf6888373c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.204ex; width:6.377ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} .\!}"></span> În consecință, mulțimea <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 \mathbb {N} \!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} \!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b6c7a4803d4cc4da7f3c4c0a534ee26a00f21d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.341ex; width:1.632ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} \!}"></span> este mărginită deci ar admite o margine superioară <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 \mathbb {R} .\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>.</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\in \mathbb {R} .\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ef34c6d35f34975356b136ee4761e0f32f99f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.204ex; width:6.07ex; height:2.176ex;" alt="{\displaystyle z\in \mathbb {R} .\!}"></span> Din definiția marginii superioare, rezultă că există <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}\in \mathbb {N} \!}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{0}\in \mathbb {N} \!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6041f96db3d548337ed0d94638af72317656940" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.341ex; width:6.921ex; height:2.509ex;" alt="{\displaystyle n_{0}\in \mathbb {N} \!}"></span> cu <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-1<n_{0}<z,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>−</mo> <mn>1</mn> <mo><</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <mi>z</mi> <mo>,</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z-1<n_{0}<z,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8bbb72f1a6874f002fb381bb8f0e819a1223053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.229ex; width:15.314ex; height:2.509ex;" alt="{\displaystyle z-1<n_{0}<z,\!}"></span> de unde avem că <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=\sup \mathbb {N} <n_{0}+1\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo><</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=\sup \mathbb {N} <n_{0}+1\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f887f23412ecabc38af19a80cfce5f2ba544109" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.218ex; width:19.134ex; height:2.509ex;" alt="{\displaystyle z=\sup \mathbb {N} <n_{0}+1\!}"></span> absurd deoarece <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 \mathbb {N} \!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} \!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b6c7a4803d4cc4da7f3c4c0a534ee26a00f21d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.341ex; width:1.632ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} \!}"></span> este inductivă (aceasta provine chiar din axiomele mulțimii <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 \mathbb {N} \!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} \!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b6c7a4803d4cc4da7f3c4c0a534ee26a00f21d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.341ex; width:1.632ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} \!}"></span>) și ca atare <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}+1\in N.\!}"> <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>1</mn> <mo>∈</mo> <mi>N</mi> <mo>.</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{0}+1\in N.\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c863a565e0de3e3e1b8537c12129551dbd2a88f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.204ex; width:11.819ex; height:2.509ex;" alt="{\displaystyle n_{0}+1\in N.\!}"></span> Așadar există un <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\in \mathbb {N} \!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\in \mathbb {N} \!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7704c06f90224050e372a8e671fa08844424fd6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.341ex; width:6.513ex; height:2.176ex;" alt="{\displaystyle m\in \mathbb {N} \!}"></span> cu <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 m.\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>m</mi> <mo>.</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq m.\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc0a1fec3a878cb7ef0dfc4433dc55dc11475250" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-right: -0.204ex; width:6.932ex; height:2.176ex;" alt="{\displaystyle x\leq m.\!}"></span> </p><p>Fie mulțimea: </p> <dl><dd><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 A=\{n\in \mathbb {N} |\;x<n\}.\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thickmathspace" /> <mi>x</mi> <mo><</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\{n\in \mathbb {N} |\;x<n\}.\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/184aaea6d5c826de969522893ab5790d000c9e20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.204ex; width:20.658ex; height:2.843ex;" alt="{\displaystyle A=\{n\in \mathbb {N} |\;x<n\}.\!}"></span></dd></dl></dd></dl> <p>Mulțimea <i>A</i> este mărginită inferior deci există <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 \mathbb {R} \!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in \mathbb {R} \!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8309fef6cd270b95b2cd17abd45d5fc51e1baef0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.348ex; width:5.635ex; height:2.509ex;" alt="{\displaystyle y\in \mathbb {R} \!}"></span> cu <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=\inf A.\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">inf</mo> <mi>A</mi> <mo>.</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\inf A.\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a88325f7790c077e436a4191f3c565950c274cd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.204ex; width:9.499ex; height:2.509ex;" alt="{\displaystyle y=\inf A.\!}"></span> Din definiția infimumului există pentru un <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 <1\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo><</mo> <mn>1</mn> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon <1\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83bddbb448d4871fbf594286a946ec0ce4be2be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.218ex; width:5.175ex; height:2.176ex;" alt="{\displaystyle \varepsilon <1\!}"></span> un <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}\in A\!}"> <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> <mi>A</mi> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{0}\in A\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b71824f14774ad651577bfb59885623ac4cece8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.331ex; width:7.623ex; height:2.509ex;" alt="{\displaystyle m_{0}\in A\!}"></span> cu <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<m_{0}<y+\varepsilon .\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo><</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <mi>y</mi> <mo>+</mo> <mi>ε</mi> <mo>.</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y<m_{0}<y+\varepsilon .\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84bfa393379f1fc2552da83c886b852f8ebf64dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.204ex; width:15.99ex; height:2.343ex;" alt="{\displaystyle y<m_{0}<y+\varepsilon .\!}"></span> Fie <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 A\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mi>A</mi> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in A\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c14c0fd73406ba9e19593fcfa3ac54043f2ac70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.331ex; width:5.923ex; height:2.176ex;" alt="{\displaystyle n\in A\!}"></span> arbitrar. Evident nu putem avea <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<y.\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo><</mo> <mi>y</mi> <mo>.</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n<y.\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51fac2c513fc912fa29669bfb912188898592f50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.204ex; width:6.112ex; height:2.176ex;" alt="{\displaystyle n<y.\!}"></span> Așadar avem fie <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<n<m_{0}<y+\varepsilon \!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo><</mo> <mi>n</mi> <mo><</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <mi>y</mi> <mo>+</mo> <mi>ε</mi> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y<n<m_{0}<y+\varepsilon \!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f5a698bedb36c5538d66ceb4d7101e870e9518b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.299ex; width:19.931ex; height:2.343ex;" alt="{\displaystyle y<n<m_{0}<y+\varepsilon \!}"></span> fie <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}<n.\!}"> <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> <mi>n</mi> <mo>.</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{0}<n.\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a7628304e138b39940aa88440c0f18cc755727e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.204ex; width:8.051ex; height:2.176ex;" alt="{\displaystyle m_{0}<n.\!}"></span> În prima situație ar rezulta că <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}-n<\varepsilon \!}"> <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> <mi>n</mi> <mo><</mo> <mi>ε</mi> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{0}-n<\varepsilon \!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08066768077aabd22ae5619ed98b387fdc246a63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.299ex; width:11.423ex; height:2.343ex;" alt="{\displaystyle m_{0}-n<\varepsilon \!}"></span> absurd. Așadar pentru orice <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 A\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mi>A</mi> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in A\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c14c0fd73406ba9e19593fcfa3ac54043f2ac70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.331ex; width:5.923ex; height:2.176ex;" alt="{\displaystyle n\in A\!}"></span> avem <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}<n\!}"> <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> <mi>n</mi> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{0}<n\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2fb378454a61551ec7cae38d080312ff9b4c04f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.341ex; width:7.541ex; height:2.176ex;" alt="{\displaystyle m_{0}<n\!}"></span> ceea ce înseamnă că <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}\inf A.\!}"> <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 movablelimits="true" form="prefix">inf</mo> <mi>A</mi> <mo>.</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{0}\inf A.\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c67b0457efce2ceb3206cfe7038a33badef9eeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.204ex; width:8.727ex; height:2.509ex;" alt="{\displaystyle m_{0}\inf A.\!}"></span> întrucât <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}\in A\!}"> <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> <mi>A</mi> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{0}\in A\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b71824f14774ad651577bfb59885623ac4cece8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.331ex; width:7.623ex; height:2.509ex;" alt="{\displaystyle m_{0}\in A\!}"></span> rezultă că <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<m_{0}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<m_{0}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15a0fe2f501a080960bb76d180a0560115601cdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:7.523ex; height:2.176ex;" alt="{\displaystyle x<m_{0}\!}"></span> iar dacă am avea <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+1<m_{0}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo><</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+1<m_{0}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c6b0ad170dd7f58174455639edc6f1f88501ef6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:11.526ex; height:2.509ex;" alt="{\displaystyle x+1<m_{0}\!}"></span> ar rezulta că <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<m_{0}-1\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<m_{0}-1\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27477a2e17311448ec2cc8e52828ec75aec4a192" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.218ex; width:11.356ex; height:2.509ex;" alt="{\displaystyle x<m_{0}-1\!}"></span> deci <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}\!}"> <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> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{0}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c923c5b85557227d95b81e683879bfab069d5954" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle m_{0}\!}"></span> nu ar mai fi <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 \inf A\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">inf</mo> <mi>A</mi> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \inf A\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04511b56643552cf2dca4d36cfa95342bc007a15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.331ex; width:4.726ex; height:2.176ex;" alt="{\displaystyle \inf A\!}"></span> absurd. Așadar avem și relația <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}<x+1.\!}"> <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> <mi>x</mi> <mo>+</mo> <mn>1.</mn> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{0}<x+1.\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add1c9b0699d92e443c757c83ec91d54e5673b9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.204ex; width:11.989ex; height:2.509ex;" alt="{\displaystyle m_{0}<x+1.\!}"></span> </p><p>Acum pentru demonstrarea proprietății lui Arhimede, se vor considera cazurile: </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 x\leq 0\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mn>0</mn> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq 0\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc8ebf567a80df102a1d0087ccf9ac7febe3e1d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-right: -0.294ex; width:5.498ex; height:2.343ex;" alt="{\displaystyle x\leq 0\!}"></span>, atunci se ia <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=1.\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1.</mn> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1.\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/791761853574b603e6ae27399b9c89152fe259d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.204ex; width:6.119ex; height:2.176ex;" alt="{\displaystyle n=1.\!}"></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 x>0.\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>></mo> <mn>0.</mn> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x>0.\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/801ae14d21567f67a79045b2caa19d23a194a8a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.204ex; width:6.054ex; height:2.176ex;" alt="{\displaystyle x>0.\!}"></span> Având în vedere că <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 xy^{-1}>0,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>></mo> <mn>0</mn> <mo>,</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy^{-1}>0,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d53f108e09c1c5ce949727bd77871a8516039049" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.229ex; width:9.573ex; height:3.009ex;" alt="{\displaystyle xy^{-1}>0,\!}"></span> putem aplica teorema precedentă.</li></ul> <p>Deci există <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 \mathbb {N} \!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} \!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9498d37575298da80cda112c0359d70b4811192a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.341ex; width:5.867ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} \!}"></span> cu <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 xy^{-1}<n\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo><</mo> <mi>n</mi> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy^{-1}<n\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53ae8313d81b564911cbc630368a2648b626c37b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.341ex; width:9.27ex; height:3.009ex;" alt="{\displaystyle xy^{-1}<n\!}"></span> de unde rezultă axioma lui Arhimede. </p> <div class="mw-heading mw-heading2"><h2 id="Legături_externe"><span id="Leg.C4.83turi_externe"></span>Legături externe</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axioma_lui_Arhimede&veaction=edit&section=3" title="Modifică secțiunea: Legături externe" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Axioma_lui_Arhimede&action=edit&section=3" title="Edit section's source code: Legături externe"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.wolframalpha.com/entities/famous_math_problems/archimedes%27_axiom/5u/6l/i6/">WolframAlpha.com</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20101111113734/http://www.wolframalpha.com/entities/famous_math_problems/archimedes%27_axiom/5u/6l/i6/">Arhivat</a> în <time datetime="2010-11-11">11 noiembrie 2010</time>, la <a href="/wiki/Wayback_Machine" class="mw-redirect" title="Wayback Machine">Wayback Machine</a>.</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; 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