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<!DOCTYPE html><html> <head> <title>limit function of sequence</title>  <meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> <link rel="stylesheet" href="LaTeXML.css" type="text/css"> <link rel="stylesheet" href="ltx-article.css" type="text/css"> <link rel="stylesheet" href="https://cdn.rawgit.com/holtzermann17/3f71ceeb3b055e1ddc3b6c11fb1f074c/raw/2bb23e3b173ff96840797fc0c3bcb8c54085df8e/LaTeXML.css" type="text/css"> <link rel="stylesheet" href="https://cdn.rawgit.com/holtzermann17/4bda0365b30858ac2fb83623185fe3ec/raw/cedd84ed3e3ad597c5d293f443ecfe4803741c6b/ltx-article.css" type="text/css"> <script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-MML-AM_CHTML" type="text/javascript"></script> </head> <body> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">limit function of sequence</h1> <div id="p1" class="ltx_para"> <br class="ltx_break"> </div> <div id="Thmthmplain1" class="ltx_theorem ltx_theorem_thmplain"> <h6 class="ltx_title ltx_runin ltx_font_bold ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Theorem 1</span>.</h6> <div id="Thmthmplain1.p1" class="ltx_para"> <p class="ltx_p">Let <math id="Thmthmplain1.p1.m1" class="ltx_Math" alttext="f_{1},\,f_{2},\,\ldots" display="inline"><mrow><msub><mi>f</mi><mn>1</mn></msub><mo rspace="4.2pt">,</mo><msub><mi>f</mi><mn>2</mn></msub><mo rspace="4.2pt">,</mo><mi mathvariant="normal">…</mi></mrow></math> be a <a class="nnexus_concept" href="http://planetmath.org/sequence">sequence</a> of <a class="nnexus_concept" href="http://mathworld.wolfram.com/RealFunction.html">real functions</a> all defined in the <a class="nnexus_concept" href="http://planetmath.org/interval">interval</a> <math id="Thmthmplain1.p1.m2" class="ltx_Math" alttext="[a,\,b]" display="inline"><mrow><mo stretchy="false">[</mo><mi>a</mi><mo rspace="4.2pt">,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow></math>. This <span class="ltx_text ltx_font_italic"><a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">function</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Function.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/function"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> sequence</span> <a class="nnexus_concept" href="http://planetmath.org/convergesuniformly">converges uniformly</a> to the <span class="ltx_text ltx_font_italic">limit function</span> <math id="Thmthmplain1.p1.m3" class="ltx_Math" alttext="f" display="inline"><mi>f</mi></math> on the interval <math id="Thmthmplain1.p1.m4" class="ltx_Math" alttext="[a,\,b]" display="inline"><mrow><mo stretchy="false">[</mo><mi>a</mi><mo rspace="4.2pt">,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow></math> if and only if</p> <table id="S0.Ex1" class="ltx_equation ltx_eqn_table"> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math id="S0.Ex1.m1" class="ltx_Math" alttext="\lim_{n\to\infty}\sup\{|f_{n}(x)-f(x)|\vdots\,\,a\leqq x\leqq b\}=0." display="block"><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo movablelimits="false">sup</mo><mrow><mo stretchy="false">{</mo><mo stretchy="false">|</mo><msub><mi>f</mi><mi>n</mi></msub><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>-</mo><mi>f</mi><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">|</mo><mpadded width="+3.3pt"><mi mathvariant="normal">⋮</mi></mpadded><mi>a</mi><mo>≦</mo><mi>x</mi><mo>≦</mo><mi>b</mi><mo stretchy="false">}</mo></mrow><mo>=</mo><mn>0</mn><mo>.</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </table> </div> </div> <div id="p2" class="ltx_para"> <p class="ltx_p">If all functions <math id="p2.m1" class="ltx_Math" alttext="f_{n}" display="inline"><msub><mi>f</mi><mi>n</mi></msub></math> are <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">continuous</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Continuous.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/continuous"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> in the interval <math id="p2.m2" class="ltx_Math" alttext="[a,\,b]" display="inline"><mrow><mo stretchy="false">[</mo><mi>a</mi><mo rspace="4.2pt">,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow></math> and <math id="p2.m3" class="ltx_Math" alttext="\lim_{n\to\infty}f_{n}(x)=f(x)" display="inline"><mrow><mrow><msub><mo>lim</mo><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></msub><mo>⁡</mo><mrow><msub><mi>f</mi><mi>n</mi></msub><mo>⁢</mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mi>f</mi><mo>⁢</mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></math> in all points <math id="p2.m4" class="ltx_Math" alttext="x" display="inline"><mi>x</mi></math> of the interval, the limit function needs not to be continuous in this interval; example <math id="p2.m5" class="ltx_Math" alttext="f_{n}(x)=\sin^{n}x" display="inline"><mrow><mrow><msub><mi>f</mi><mi>n</mi></msub><mo>⁢</mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi>sin</mi><mi>n</mi></msup><mo>⁡</mo><mi>x</mi></mrow></mrow></math> in <math id="p2.m6" class="ltx_Math" alttext="[0,\,\pi]" display="inline"><mrow><mo stretchy="false">[</mo><mn>0</mn><mo rspace="4.2pt">,</mo><mi>π</mi><mo stretchy="false">]</mo></mrow></math>:</p> <img src="pmplot" id="p2.g1" class="ltx_graphics ltx_centering" alt=""> </div> <div id="Thmthmplain2" class="ltx_theorem ltx_theorem_thmplain"> <h6 class="ltx_title ltx_runin ltx_font_bold ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Theorem 2</span>.</h6> <div id="Thmthmplain2.p1" class="ltx_para"> <p class="ltx_p">If all the functions <math id="Thmthmplain2.p1.m1" class="ltx_Math" alttext="f_{n}" display="inline"><msub><mi>f</mi><mi>n</mi></msub></math> are continuous and the sequence <math id="Thmthmplain2.p1.m2" class="ltx_Math" alttext="f_{1},\,f_{2},\,\ldots" display="inline"><mrow><msub><mi>f</mi><mn>1</mn></msub><mo rspace="4.2pt">,</mo><msub><mi>f</mi><mn>2</mn></msub><mo rspace="4.2pt">,</mo><mi mathvariant="normal">…</mi></mrow></math> converges uniformly to a function <math id="Thmthmplain2.p1.m3" class="ltx_Math" alttext="f" display="inline"><mi>f</mi></math> in the interval <math id="Thmthmplain2.p1.m4" class="ltx_Math" alttext="[a,\,b]" display="inline"><mrow><mo stretchy="false">[</mo><mi>a</mi><mo rspace="4.2pt">,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow></math>, then the limit function <math id="Thmthmplain2.p1.m5" class="ltx_Math" alttext="f" display="inline"><mi>f</mi></math> is continuous in this interval.</p> </div> </div> <div id="p3" class="ltx_para"> <p class="ltx_p"><span class="ltx_text ltx_font_bold">Note.</span> The notion of can be extended to the sequences of complex functions (the interval is replaced with some subset <math id="p3.m1" class="ltx_Math" alttext="G" display="inline"><mi>G</mi></math> of <math id="p3.m2" class="ltx_Math" alttext="\mathbb{C}" display="inline"><mi>ℂ</mi></math>). The limit function of a <a class="nnexus_concept" href="http://planetmath.org/uniformconvergence">uniformly convergent</a> sequence of <a class="nnexus_concept" href="http://mathworld.wolfram.com/ContinuousFunction.html">continuous functions</a> is continuous in <math id="p3.m3" class="ltx_Math" alttext="G" display="inline"><mi>G</mi></math>. </p> <table class="ltx_tabular ltx_align_right ltx_guessed_headers ltx_align_middle"> <tbody class="ltx_tbody"> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">Title</th> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">limit function of sequence</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Canonical name</th> <td class="ltx_td ltx_align_left ltx_border_r">LimitFunctionOfSequence</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Date of creation</th> <td class="ltx_td ltx_align_left ltx_border_r">2013-03-22 14:37:45</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Last modified on</th> <td class="ltx_td ltx_align_left ltx_border_r">2013-03-22 14:37:45</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Owner</th> <td class="ltx_td ltx_align_left ltx_border_r">pahio (2872)</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Last modified by</th> <td class="ltx_td ltx_align_left ltx_border_r">pahio (2872)</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Numerical id</th> <td class="ltx_td ltx_align_left ltx_border_r">22</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Author</th> <td class="ltx_td ltx_align_left ltx_border_r">pahio (2872)</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Entry type</th> <td class="ltx_td ltx_align_left ltx_border_r">Theorem</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Classification</th> <td class="ltx_td ltx_align_left ltx_border_r">msc 26A15</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Classification</th> <td class="ltx_td ltx_align_left ltx_border_r">msc 40A30</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Related topic</th> <td class="ltx_td ltx_align_left ltx_border_r">LimitOfAUniformlyConvergentSequenceOfContinuousFunctionsIsContinuous</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Related topic</th> <td class="ltx_td ltx_align_left ltx_border_r">PointPreventingUniformConvergence</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Defines</th> <td class="ltx_td ltx_align_left ltx_border_r">function sequence</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">Defines</th> <td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">limit function</td> </tr> </tbody> </table> </div> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Fri Feb 9 19:48:45 2018 by <a href="http://dlmf.nist.gov/LaTeXML/">LaTeXML <img src="data:image/png;base64,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" alt="[LOGO]"></a> </div></footer> </div> </body> </html>