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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</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">(Top)</div> </a> </li> <li id="toc-Forms_of_completeness" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Forms_of_completeness"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Forms of completeness</span> </div> </a> <button aria-controls="toc-Forms_of_completeness-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Forms of completeness subsection</span> </button> <ul id="toc-Forms_of_completeness-sublist" class="vector-toc-list"> <li id="toc-Least_upper_bound_property" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Least_upper_bound_property"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Least upper bound property</span> </div> </a> <ul id="toc-Least_upper_bound_property-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dedekind_completeness" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dedekind_completeness"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Dedekind completeness</span> </div> </a> <ul id="toc-Dedekind_completeness-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cauchy_completeness" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cauchy_completeness"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Cauchy completeness</span> </div> </a> <ul id="toc-Cauchy_completeness-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nested_intervals_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nested_intervals_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Nested intervals theorem</span> </div> </a> <ul id="toc-Nested_intervals_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_open_induction_principle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_open_induction_principle"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>The open induction principle</span> </div> </a> <ul id="toc-The_open_induction_principle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Monotone_convergence_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Monotone_convergence_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.6</span> <span>Monotone convergence theorem</span> </div> </a> <ul id="toc-Monotone_convergence_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bolzano–Weierstrass_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bolzano–Weierstrass_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.7</span> <span>Bolzano–Weierstrass theorem</span> </div> </a> <ul id="toc-Bolzano–Weierstrass_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_intermediate_value_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_intermediate_value_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.8</span> <span>The intermediate value theorem</span> </div> </a> <ul id="toc-The_intermediate_value_theorem-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> </ul> 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class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Axioma_del_supremo" title="Axioma del supremo – Spanish" lang="es" hreflang="es" data-title="Axioma del supremo" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D9%85%D8%A7%D9%85%DB%8C%D8%AA_%D8%A7%D8%B9%D8%AF%D8%A7%D8%AF_%D8%AD%D9%82%DB%8C%D9%82%DB%8C" title="تمامیت اعداد حقیقی – Persian" lang="fa" hreflang="fa" data-title="تمامیت اعداد حقیقی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%8B%A4%EC%88%98%EC%9D%98_%EC%99%84%EB%B9%84%EC%84%B1" title="실수의 완비성 – Korean" lang="ko" hreflang="ko" data-title="실수의 완비성" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Assioma_di_Dedekind" title="Assioma di Dedekind – Italian" lang="it" hreflang="it" data-title="Assioma di Dedekind" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Assioma_de_Dedekind" title="Assioma de Dedekind – Lombard" lang="lmo" hreflang="lmo" data-title="Assioma de Dedekind" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%AE%9F%E6%95%B0%E3%81%AE%E9%80%A3%E7%B6%9A%E6%80%A7" title="実数の連続性 – Japanese" lang="ja" hreflang="ja" data-title="実数の連続性" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Aksjomat_ci%C4%85g%C5%82o%C5%9Bci" title="Aksjomat ciągłości – Polish" lang="pl" hreflang="pl" data-title="Aksjomat ciągłości" data-language-autonym="Polski" data-language-local-name="Polish" 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/Axioma_do_supremo" title="Axioma do supremo – Portuguese" lang="pt" hreflang="pt" data-title="Axioma do supremo" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li 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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">From Wikipedia, the free encyclopedia</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="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Concept in mathematics</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Completeness_(logic)" title="Completeness (logic)">Completeness (logic)</a>.</div> <p> <b>Completeness</b> is a property of the <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a> that, intuitively, implies that there are no "gaps" (in Dedekind's terminology) or "missing points" in the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real number line</a>. This contrasts with the <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>, whose corresponding number line has a "gap" at each <a href="/wiki/Irrational_number" title="Irrational number">irrational</a> value. In the <a href="/wiki/Decimal" title="Decimal">decimal number system</a>, completeness is equivalent to the statement that any infinite string of decimal digits is actually a <a href="/wiki/Decimal_representation" title="Decimal representation">decimal representation</a> for some real number. </p><p>Depending on the <a href="/wiki/Construction_of_the_real_numbers" title="Construction of the real numbers">construction of the real numbers</a> used, completeness may take the form of an <a href="/wiki/Axiom" title="Axiom">axiom</a> (the <b>completeness axiom</b>), or may be a <a href="/wiki/Theorem" title="Theorem">theorem</a> proven from the construction. There are many <a href="/wiki/Logical_equivalence" title="Logical equivalence">equivalent</a> forms of completeness, the most prominent being <b>Dedekind completeness</b> and <b>Cauchy completeness</b> (<a href="/wiki/Complete_metric_space" title="Complete metric space">completeness as a metric space</a>). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Forms_of_completeness">Forms of completeness</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Completeness_of_the_real_numbers&action=edit&section=1" title="Edit section: Forms of completeness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Real_number" title="Real number">real numbers</a> can be <a href="/wiki/Construction_of_the_real_numbers#Synthetic_approach" title="Construction of the real numbers">defined synthetically</a> as an <a href="/wiki/Ordered_field" title="Ordered field">ordered field</a> satisfying some version of the <i>completeness axiom</i>. Different versions of this axiom are all equivalent in the sense that any ordered field that satisfies one form of completeness satisfies all of them, apart from Cauchy completeness and nested intervals theorem, which are strictly weaker in that there are <a href="/wiki/Non-Archimedean_ordered_field" title="Non-Archimedean ordered field">non</a> <a href="/wiki/Archimedean_property" title="Archimedean property">Archimedean fields</a> that are ordered and Cauchy complete. When the real numbers are instead constructed using a model, completeness becomes a <a href="/wiki/Theorem" title="Theorem">theorem</a> or collection of theorems. </p> <div class="mw-heading mw-heading3"><h3 id="Least_upper_bound_property">Least upper bound property</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Completeness_of_the_real_numbers&action=edit&section=2" title="Edit section: Least upper bound property"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Least-upper-bound_property" title="Least-upper-bound property">Least-upper-bound property</a></div> <p>The <b>least-upper-bound property</b> states that every <a href="/wiki/Nonempty" class="mw-redirect" title="Nonempty">nonempty</a> subset of real numbers having an <a href="/wiki/Upper_bound" class="mw-redirect" title="Upper bound">upper bound</a> (or bounded above) must have a <a href="/wiki/Least_upper_bound" class="mw-redirect" title="Least upper bound">least upper bound</a> (or supremum) in the set of real numbers. </p><p>The <a href="/wiki/Rational_number" title="Rational number">rational number line</a> <b>Q</b> does not have the least upper bound property. An example is the subset of rational numbers </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\{x\in \mathbb {Q} \mid x^{2}<2\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>∣</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo><</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\{x\in \mathbb {Q} \mid x^{2}<2\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8224e87681e9892da8ea1a842bed05a8cafd5972" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.13ex; height:3.176ex;" alt="{\displaystyle S=\{x\in \mathbb {Q} \mid x^{2}<2\}.}"></span></dd></dl> <p>This set has an upper bound. However, this set has no least upper bound in <span class="texhtml"><b>Q</b></span>: the least upper bound as a subset of the reals would be <span class="texhtml">√2</span>, but it does not exist in <span class="texhtml"><b>Q</b></span>. For any upper bound <span class="texhtml"><i>x</i> ∈ <b>Q</b></span>, there is another upper bound <span class="texhtml"><i>y</i> ∈ <b>Q</b></span> with <span class="texhtml"><i>y</i> < <i>x</i></span>. </p><p>For instance, take <span class="texhtml"><i>x</i> = 1.5</span>, then <span class="texhtml mvar" style="font-style:italic;">x</span> is certainly an upper bound of <span class="texhtml mvar" style="font-style:italic;">S</span>, since <span class="texhtml mvar" style="font-style:italic;">x</span> is positive and <span class="texhtml"><i>x</i><sup>2</sup> = 2.25 ≥ 2</span>; that is, no element of <span class="texhtml mvar" style="font-style:italic;">S</span> is larger than <span class="texhtml mvar" style="font-style:italic;">x</span>. However, we can choose a smaller upper bound, say <span class="texhtml"><i>y</i> = 1.45</span>; this is also an upper bound of <span class="texhtml mvar" style="font-style:italic;">S</span> for the same reasons, but it is smaller than <span class="texhtml mvar" style="font-style:italic;">x</span>, so <span class="texhtml mvar" style="font-style:italic;">x</span> is not a least-upper-bound of <span class="texhtml mvar" style="font-style:italic;">S</span>. We can proceed similarly to find an upper bound of <span class="texhtml mvar" style="font-style:italic;">S</span> that is smaller than <span class="texhtml mvar" style="font-style:italic;">y</span>, say <span class="texhtml"><i>z</i> = 1.42</span>, etc., such that we never find a least-upper-bound of <span class="texhtml mvar" style="font-style:italic;">S</span> in <span class="texhtml"><b>Q</b></span>. </p><p>The least upper bound property can be generalized to the setting of <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered sets</a>. See <a href="/wiki/Completeness_(order_theory)" title="Completeness (order theory)">completeness (order theory)</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Dedekind_completeness">Dedekind completeness</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Completeness_of_the_real_numbers&action=edit&section=3" title="Edit section: Dedekind completeness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><i>See <a href="/wiki/Dedekind_completeness" class="mw-redirect" title="Dedekind completeness">Dedekind completeness</a> for more general concepts bearing this name.</i></dd></dl> <p>Dedekind completeness is the property that every <a href="/wiki/Dedekind_cut" title="Dedekind cut">Dedekind cut</a> of the real numbers is generated by a real number. In a synthetic approach to the real numbers, this is the version of completeness that is most often included as an axiom. </p><p>The <a href="/wiki/Rational_number" title="Rational number">rational number line</a> <b>Q</b> is not Dedekind complete. An example is the Dedekind cut </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=\{x\in \mathbb {Q} \mid x^{2}\leq 2\vee x<0\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>∣</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≤</mo> <mn>2</mn> <mo>∨</mo> <mi>x</mi> <mo><</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=\{x\in \mathbb {Q} \mid x^{2}\leq 2\vee x<0\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73df7bd6cffb054e0277d1596897fe67f562af78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.387ex; height:3.176ex;" alt="{\displaystyle L=\{x\in \mathbb {Q} \mid x^{2}\leq 2\vee x<0\}.}"></span></dd> <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 R=\{x\in \mathbb {Q} \mid x^{2}\geq 2\wedge x>0\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>∣</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≥</mo> <mn>2</mn> <mo>∧</mo> <mi>x</mi> <mo>></mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=\{x\in \mathbb {Q} \mid x^{2}\geq 2\wedge x>0\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b941127320672fe394dfbeff73dddcd701251392" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.568ex; height:3.176ex;" alt="{\displaystyle R=\{x\in \mathbb {Q} \mid x^{2}\geq 2\wedge x>0\}.}"></span></dd></dl> <p><i>L</i> does not have a maximum and <i>R</i> does not have a minimum, so this cut is not generated by a rational number. </p><p>There is a <a href="/wiki/Construction_of_the_real_numbers#Construction_by_Dedekind_cuts" title="Construction of the real numbers">construction of the real numbers</a> based on the idea of using Dedekind cuts of rational numbers to name real numbers; e.g. the cut <i>(L,R)</i> described above would name <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 {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span>. If one were to repeat the construction of real numbers with Dedekind cuts (i.e., "close" the set of real numbers by adding all possible Dedekind cuts), one would obtain no additional numbers because the real numbers are already Dedekind complete. </p> <div class="mw-heading mw-heading3"><h3 id="Cauchy_completeness">Cauchy completeness</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Completeness_of_the_real_numbers&action=edit&section=4" title="Edit section: Cauchy completeness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Cauchy completeness</b> is the statement that every <a href="/wiki/Cauchy_sequence" title="Cauchy sequence">Cauchy sequence</a> of real numbers <a href="/wiki/Convergent_sequences" class="mw-redirect" title="Convergent sequences">converges</a> to a real number. </p><p>The <a href="/wiki/Rational_number" title="Rational number">rational number line</a> <b>Q</b> is not Cauchy complete. An example is the following sequence of rational numbers: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3,\quad 3.1,\quad 3.14,\quad 3.142,\quad 3.1416,\quad \ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>,</mo> <mspace width="1em" /> <mn>3.1</mn> <mo>,</mo> <mspace width="1em" /> <mn>3.14</mn> <mo>,</mo> <mspace width="1em" /> <mn>3.142</mn> <mo>,</mo> <mspace width="1em" /> <mn>3.1416</mn> <mo>,</mo> <mspace width="1em" /> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3,\quad 3.1,\quad 3.14,\quad 3.142,\quad 3.1416,\quad \ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6018fe7b060bb29c8927e99a5d2618b6c6f2573e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:39.53ex; height:2.509ex;" alt="{\displaystyle 3,\quad 3.1,\quad 3.14,\quad 3.142,\quad 3.1416,\quad \ldots }"></span></dd></dl> <p>Here the <i>n</i>th term in the sequence is the <i>n</i>th decimal approximation for <a href="/wiki/Pi" title="Pi">pi</a>. Though this is a Cauchy sequence of rational numbers, it does not converge to any rational number. (In this real number line, this sequence converges to pi.) </p><p>Cauchy completeness is related to the construction of the real numbers using Cauchy sequences. Essentially, this method defines a real number to be the limit of a Cauchy sequence of rational numbers. </p><p>In <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, Cauchy completeness can be generalized to a notion of completeness for any <a href="/wiki/Metric_space" title="Metric space">metric space</a>. See <a href="/wiki/Complete_metric_space" title="Complete metric space">complete metric space</a>. </p><p>For an <a href="/wiki/Ordered_field" title="Ordered field">ordered field</a>, Cauchy completeness is weaker than the other forms of completeness on this page. But Cauchy completeness and the <a href="/wiki/Archimedean_property" title="Archimedean property">Archimedean property</a> taken together are equivalent to the others. </p> <div class="mw-heading mw-heading3"><h3 id="Nested_intervals_theorem">Nested intervals theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Completeness_of_the_real_numbers&action=edit&section=5" title="Edit section: Nested intervals theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Nested_intervals" title="Nested intervals">Nested intervals</a></div> <p>The <b>nested interval theorem</b> is another form of completeness. Let <span class="texhtml"><i>I<sub>n</sub></i> = [<i>a<sub>n</sub></i>, <i>b<sub>n</sub></i>]</span> be a sequence of closed <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">intervals</a>, and suppose that these intervals are nested in the sense that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{1}\;\supset \;I_{2}\;\supset \;I_{3}\;\supset \;\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thickmathspace" /> <mo>⊃</mo> <mspace width="thickmathspace" /> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thickmathspace" /> <mo>⊃</mo> <mspace width="thickmathspace" /> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mspace width="thickmathspace" /> <mo>⊃</mo> <mspace width="thickmathspace" /> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{1}\;\supset \;I_{2}\;\supset \;I_{3}\;\supset \;\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ef3a06baedf410f7006f82c393f9b1aadc03758" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.122ex; height:2.509ex;" alt="{\displaystyle I_{1}\;\supset \;I_{2}\;\supset \;I_{3}\;\supset \;\cdots }"></span></dd></dl> <p>Moreover, assume that <span class="texhtml"><i>b<sub>n</sub></i> − <i>a<sub>n</sub></i> → 0</span> as <span class="texhtml"><i>n</i> → +∞</span>. The nested interval theorem states that the <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a> of all of the intervals <span class="texhtml"><i>I<sub>n</sub></i></span> contains exactly one point. </p><p>The <a href="/wiki/Rational_number" title="Rational number">rational number line</a> does not satisfy the nested interval theorem. For example, the sequence (whose terms are derived from the digits of <a href="/wiki/Pi" title="Pi">pi</a> in the suggested way) </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [3,4]\;\supset \;[3.1,3.2]\;\supset \;[3.14,3.15]\;\supset \;[3.141,3.142]\;\supset \;\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo stretchy="false">]</mo> <mspace width="thickmathspace" /> <mo>⊃</mo> <mspace width="thickmathspace" /> <mo stretchy="false">[</mo> <mn>3.1</mn> <mo>,</mo> <mn>3.2</mn> <mo stretchy="false">]</mo> <mspace width="thickmathspace" /> <mo>⊃</mo> <mspace width="thickmathspace" /> <mo stretchy="false">[</mo> <mn>3.14</mn> <mo>,</mo> <mn>3.15</mn> <mo stretchy="false">]</mo> <mspace width="thickmathspace" /> <mo>⊃</mo> <mspace width="thickmathspace" /> <mo stretchy="false">[</mo> <mn>3.141</mn> <mo>,</mo> <mn>3.142</mn> <mo stretchy="false">]</mo> <mspace width="thickmathspace" /> <mo>⊃</mo> <mspace width="thickmathspace" /> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [3,4]\;\supset \;[3.1,3.2]\;\supset \;[3.14,3.15]\;\supset \;[3.141,3.142]\;\supset \;\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ab26179b0bef3009146785a269f6a229f148345" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.719ex; height:2.843ex;" alt="{\displaystyle [3,4]\;\supset \;[3.1,3.2]\;\supset \;[3.14,3.15]\;\supset \;[3.141,3.142]\;\supset \;\cdots }"></span></dd></dl> <p>is a nested sequence of closed intervals in the rational numbers whose intersection is empty. (In the real numbers, the intersection of these intervals contains the number <a href="/wiki/Pi" title="Pi">pi</a>.) </p><p>Nested intervals theorem shares the same logical status as Cauchy completeness in this spectrum of expressions of completeness. In other words, nested intervals theorem by itself is weaker than other forms of completeness, although taken together with <a href="/wiki/Archimedean_property" title="Archimedean property">Archimedean property</a>, it is equivalent to the others. </p> <div class="mw-heading mw-heading3"><h3 id="The_open_induction_principle">The open induction principle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Completeness_of_the_real_numbers&action=edit&section=6" title="Edit section: The open induction principle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The open induction principle states that a non-empty open subset <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> of the interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span> must be equal to the entire interval, if for any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\in [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\in [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d206089575b085a14500241de3977673eaa92881" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.444ex; height:2.843ex;" alt="{\displaystyle r\in [a,b]}"></span>, we have that <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,r)\subset S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>⊂</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,r)\subset S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/994c0497e4b78608e439ca056b66e061a20a0033" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.462ex; height:2.843ex;" alt="{\displaystyle [a,r)\subset S}"></span> implies <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,r]\subset S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">]</mo> <mo>⊂</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,r]\subset S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6311dfedf3aa2cabec651b22bf4350e03316a762" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.204ex; height:2.843ex;" alt="{\displaystyle [a,r]\subset S}"></span>. </p><p>The open induction principle can be shown to be equivalent to Dedekind completeness for arbitrary ordered sets under the order topology, using proofs by contradiction. In weaker foundations such as in <a href="/wiki/Constructive_analysis" title="Constructive analysis">constructive analysis</a> where the law of the excluded middle does not hold, the full form of the least upper bound property fails for the Dedekind reals, while the open induction property remains true in most models (following from Brouwer's bar theorem) and is strong enough to give short proofs of key theorems. </p> <div class="mw-heading mw-heading3"><h3 id="Monotone_convergence_theorem">Monotone convergence theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Completeness_of_the_real_numbers&action=edit&section=7" title="Edit section: Monotone convergence theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b><a href="/wiki/Monotone_convergence_theorem#Convergence_of_a_monotone_sequence_of_real_numbers" title="Monotone convergence theorem">monotone convergence theorem</a></b> (described as the <b>fundamental axiom of analysis</b> by Körner<sup id="cite_ref-Körner2004_1-0" class="reference"><a href="#cite_note-Körner2004-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>) states that every nondecreasing, bounded sequence of real numbers converges. This can be viewed as a special case of the least upper bound property, but it can also be used fairly directly to prove the Cauchy completeness of the real numbers. </p> <div class="mw-heading mw-heading3"><h3 id="Bolzano–Weierstrass_theorem"><span id="Bolzano.E2.80.93Weierstrass_theorem"></span>Bolzano–Weierstrass theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Completeness_of_the_real_numbers&action=edit&section=8" title="Edit section: Bolzano–Weierstrass theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b><a href="/wiki/Bolzano%E2%80%93Weierstrass_theorem" title="Bolzano–Weierstrass theorem">Bolzano–Weierstrass theorem</a></b> states that every bounded sequence of real numbers has a convergent <a href="/wiki/Subsequence" title="Subsequence">subsequence</a>. Again, this theorem is equivalent to the other forms of completeness given above. </p> <div class="mw-heading mw-heading3"><h3 id="The_intermediate_value_theorem">The intermediate value theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Completeness_of_the_real_numbers&action=edit&section=9" title="Edit section: The intermediate value theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b><a href="/wiki/Intermediate_value_theorem" title="Intermediate value theorem">intermediate value theorem</a></b> states that every continuous function that attains both negative and positive values has a root. This is a consequence of the least upper bound property, but it can also be used to prove the least upper bound property if treated as an axiom. (The definition of continuity does not depend on any form of completeness, so there is no circularity: what is meant is that the intermediate value theorem and the least upper bound property are equivalent statements.) </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Completeness_of_the_real_numbers&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/List_of_real_analysis_topics" title="List of real analysis topics">List of real analysis topics</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Completeness_of_the_real_numbers&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Körner2004-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Körner2004_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFKörner2004" class="citation book cs1"><a href="/wiki/Thomas_William_K%C3%B6rner" title="Thomas William Körner">Körner, Thomas William</a> (2004). <i>A companion to analysis: a second first and first second course in analysis</i>. AMS Chelsea. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780821834473" title="Special:BookSources/9780821834473"><bdi>9780821834473</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+companion+to+analysis%3A+a+second+first+and+first+second+course+in+analysis&rft.pub=AMS+Chelsea&rft.date=2004&rft.isbn=9780821834473&rft.aulast=K%C3%B6rner&rft.aufirst=Thomas+William&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACompleteness+of+the+real+numbers" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Completeness_of_the_real_numbers&action=edit&section=12" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAliprantisBurkinshaw1998" class="citation book cs1"><a href="/wiki/Charalambos_D._Aliprantis" class="mw-redirect" title="Charalambos D. Aliprantis">Aliprantis, Charalambos D.</a>; Burkinshaw, Owen (1998). <i>Principles of real analysis</i> (3rd ed.). Academic. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-050257-7" title="Special:BookSources/0-12-050257-7"><bdi>0-12-050257-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+real+analysis&rft.edition=3rd&rft.pub=Academic&rft.date=1998&rft.isbn=0-12-050257-7&rft.aulast=Aliprantis&rft.aufirst=Charalambos+D.&rft.au=Burkinshaw%2C+Owen&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACompleteness+of+the+real+numbers" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrowder,_Andrew1996" class="citation book cs1">Browder, Andrew (1996). <i>Mathematical Analysis: An Introduction</i>. <a href="/wiki/Undergraduate_Texts_in_Mathematics" title="Undergraduate Texts in Mathematics">Undergraduate Texts in Mathematics</a>. New York City: Springer Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94614-4" title="Special:BookSources/0-387-94614-4"><bdi>0-387-94614-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Analysis%3A+An+Introduction&rft.place=New+York+City&rft.series=Undergraduate+Texts+in+Mathematics&rft.pub=Springer+Verlag&rft.date=1996&rft.isbn=0-387-94614-4&rft.au=Browder%2C+Andrew&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACompleteness+of+the+real+numbers" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBartle,_Robert_G.Sherbert,_Donald_R.2000" class="citation book cs1">Bartle, Robert G.; Sherbert, Donald R. (2000). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontore00bart_1"><i>Introduction to Real Analysis</i></a></span> (3rd ed.). New York City: John Wiley and Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-32148-6" title="Special:BookSources/0-471-32148-6"><bdi>0-471-32148-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Real+Analysis&rft.place=New+York+City&rft.edition=3rd&rft.pub=John+Wiley+and+Sons&rft.date=2000&rft.isbn=0-471-32148-6&rft.au=Bartle%2C+Robert+G.&rft.au=Sherbert%2C+Donald+R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontore00bart_1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACompleteness+of+the+real+numbers" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbbott,_Stephen2001" class="citation book cs1">Abbott, Stephen (2001). <i>Understanding Analysis</i>. Undergraduate Texts in Mathematics. New York: Springer Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-95060-5" title="Special:BookSources/0-387-95060-5"><bdi>0-387-95060-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Understanding+Analysis&rft.place=New+York&rft.series=Undergraduate+Texts+in+Mathematics&rft.pub=Springer+Verlag&rft.date=2001&rft.isbn=0-387-95060-5&rft.au=Abbott%2C+Stephen&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACompleteness+of+the+real+numbers" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRudin,_Walter1976" class="citation book cs1">Rudin, Walter (1976). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/principlesofmath00rudi"><i>Principles of Mathematical Analysis</i></a></span>. Walter Rudin Student Series in Advanced Mathematics (3rd ed.). McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780070542358" title="Special:BookSources/9780070542358"><bdi>9780070542358</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Mathematical+Analysis&rft.series=Walter+Rudin+Student+Series+in+Advanced+Mathematics&rft.edition=3rd&rft.pub=McGraw-Hill&rft.date=1976&rft.isbn=9780070542358&rft.au=Rudin%2C+Walter&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fprinciplesofmath00rudi&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACompleteness+of+the+real+numbers" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDangello,_FrankSeyfried,_Michael1999" class="citation book cs1">Dangello, Frank; Seyfried, Michael (1999). <i>Introductory Real Analysis</i>. Brooks Cole. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780395959336" title="Special:BookSources/9780395959336"><bdi>9780395959336</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introductory+Real+Analysis&rft.pub=Brooks+Cole&rft.date=1999&rft.isbn=9780395959336&rft.au=Dangello%2C+Frank&rft.au=Seyfried%2C+Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACompleteness+of+the+real+numbers" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBressoud,_David2007" class="citation book cs1">Bressoud, David (2007). <i>A Radical Approach to Real Analysis</i>. <a href="/wiki/Mathematical_Association_of_America" title="Mathematical Association of America">MAA</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-747-2" title="Special:BookSources/978-0-88385-747-2"><bdi>978-0-88385-747-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Radical+Approach+to+Real+Analysis&rft.pub=MAA&rft.date=2007&rft.isbn=978-0-88385-747-2&rft.au=Bressoud%2C+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACompleteness+of+the+real+numbers" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.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 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