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<!DOCTYPE html><html> <head> <title>representation of real numbers</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">representation of real numbers</h1> <div id="p1" class="ltx_para"> <br class="ltx_break"> </div> <section id="S0.SS1" class="ltx_subsection"> <h2 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">0.1 </span>Introduction</h2> <div id="S0.SS1.p1" class="ltx_para"> <p class="ltx_p">It is well-known that there are several methods to introduce the <span class="ltx_text ltx_font_italic"><a class="nnexus_concept" href="http://planetmath.org/realnumber">real numbers</a></span>. We shall follow an inductive method which is instructive as well as <a class="nnexus_concept" href="http://planetmath.org/elementaryrecursivefunction">elementary</a>. Apart from that such treatment is modern, interesting and is obtained through two <a class="nnexus_concept" href="http://mathworld.wolfram.com/Theorem.html">theorems</a> and a lemma, which are relatively easy to understand. So that our aim will be to prove the above <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">propositions</a><sup style="display: none;"><a class="nnexus_concept" href="http://planetmath.org/booleanvaluedfunction"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/propositionallogic"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup>. Our starting point is the following theorem.</p> </div> <div id="Thmtheorem1" class="ltx_theorem ltx_theorem_theorem"> <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="Thmtheorem1.p1" class="ltx_para"> <p class="ltx_p"><span class="ltx_text ltx_font_italic">Let <math id="Thmtheorem1.p1.m1" class="ltx_Math" alttext="\{a_{i}\}" display="inline"><mrow><mo mathvariant="normal" stretchy="false">{</mo><msub><mi>a</mi><mi>i</mi></msub><mo mathvariant="normal" stretchy="false">}</mo></mrow></math> be a <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">sequence</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Sequence.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/sequence"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> of <a class="nnexus_concept" href="http://mathworld.wolfram.com/PositiveInteger.html">positive integers</a> such that <math id="Thmtheorem1.p1.m2" class="ltx_Math" alttext="a_{i}>1" display="inline"><mrow><msub><mi>a</mi><mi>i</mi></msub><mo mathvariant="normal">></mo><mn mathvariant="normal">1</mn></mrow></math>, for all <math id="Thmtheorem1.p1.m3" class="ltx_Math" alttext="i\geq 1" display="inline"><mrow><mi>i</mi><mo mathvariant="normal">≥</mo><mn mathvariant="normal">1</mn></mrow></math>. Then any real number <math id="Thmtheorem1.p1.m4" class="ltx_Math" alttext="\rho" display="inline"><mi>ρ</mi></math> is uniquely expressible by</span></p> <table id="S0.E1" 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.E1.m1" class="ltx_Math" alttext="\rho=b_{0}+\sum_{i=1}^{\infty}\frac{b_{i}}{\prod_{j=1}^{i}a_{j}}," display="block"><mrow><mrow><mi>ρ</mi><mo>=</mo><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msub><mi>b</mi><mi>i</mi></msub><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mi>j</mi></msub></mrow></mfrac></mrow></mrow></mrow><mo>,</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr> </table> <p class="ltx_p"><span class="ltx_text ltx_font_italic">where the <math id="Thmtheorem1.p1.m5" class="ltx_Math" alttext="b_{i}" display="inline"><msub><mi>b</mi><mi>i</mi></msub></math> are <a class="nnexus_concept" href="http://mathworld.wolfram.com/Integer.html">integers</a> satisfying the <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">inequalities</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Inequality.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/inequalitiesforrealnumbers"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> <math id="Thmtheorem1.p1.m6" class="ltx_Math" alttext="0\leq b_{i}\leq a_{i}-1" display="inline"><mrow><mn mathvariant="normal">0</mn><mo mathvariant="normal">≤</mo><msub><mi>b</mi><mi>i</mi></msub><mo mathvariant="normal">≤</mo><mrow><msub><mi>a</mi><mi>i</mi></msub><mo mathvariant="normal">-</mo><mn mathvariant="normal">1</mn></mrow></mrow></math> for all <math id="Thmtheorem1.p1.m7" class="ltx_Math" alttext="i\geq 1" display="inline"><mrow><mi>i</mi><mo mathvariant="normal">≥</mo><mn mathvariant="normal">1</mn></mrow></math>, and <math id="Thmtheorem1.p1.m8" class="ltx_Math" alttext="b_{i}<a_{i}-1" display="inline"><mrow><msub><mi>b</mi><mi>i</mi></msub><mo mathvariant="normal"><</mo><mrow><msub><mi>a</mi><mi>i</mi></msub><mo mathvariant="normal">-</mo><mn mathvariant="normal">1</mn></mrow></mrow></math> for infinitely many <math id="Thmtheorem1.p1.m9" class="ltx_Math" alttext="i" display="inline"><mi>i</mi></math>.</span></p> </div> </div> <div class="ltx_proof"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div id="S0.SS1.p2" class="ltx_para"> <p class="ltx_p">Let <math id="S0.SS1.p2.m1" class="ltx_Math" alttext="\{b_{i}\}_{0}^{\infty}" display="inline"><msubsup><mrow><mo stretchy="false">{</mo><msub><mi>b</mi><mi>i</mi></msub><mo stretchy="false">}</mo></mrow><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup></math> be a sequence of integers and <math id="S0.SS1.p2.m2" class="ltx_Math" alttext="\{\rho_{i}\}_{1}^{\infty}" display="inline"><msubsup><mrow><mo stretchy="false">{</mo><msub><mi>ρ</mi><mi>i</mi></msub><mo stretchy="false">}</mo></mrow><mn>1</mn><mi mathvariant="normal">∞</mi></msubsup></math> a sequence of real numbers defined by the equations</p> <table id="S0.E2" 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.E2.m1" class="ltx_Math" alttext="b_{0}=[\rho],\qquad\rho_{1}=\rho-b_{0}," display="block"><mrow><mrow><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>=</mo><mrow><mo stretchy="false">[</mo><mi>ρ</mi><mo stretchy="false">]</mo></mrow></mrow><mo rspace="22.5pt">,</mo><mrow><msub><mi>ρ</mi><mn>1</mn></msub><mo>=</mo><mrow><mi>ρ</mi><mo>-</mo><msub><mi>b</mi><mn>0</mn></msub></mrow></mrow></mrow><mo>,</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr> </table> <p class="ltx_p">and for all <math id="S0.SS1.p2.m3" class="ltx_Math" alttext="i\geq 1" display="inline"><mrow><mi>i</mi><mo>≥</mo><mn>1</mn></mrow></math></p> <table id="S0.E3" 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.E3.m1" class="ltx_Math" alttext="b_{i}=[a_{i}\rho_{i}],\qquad\rho_{i+1}=a_{i}\rho_{i}-b_{i}," display="block"><mrow><mrow><mrow><msub><mi>b</mi><mi>i</mi></msub><mo>=</mo><mrow><mo stretchy="false">[</mo><mrow><msub><mi>a</mi><mi>i</mi></msub><mo>⁢</mo><msub><mi>ρ</mi><mi>i</mi></msub></mrow><mo stretchy="false">]</mo></mrow></mrow><mo rspace="22.5pt">,</mo><mrow><msub><mi>ρ</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mrow><mrow><msub><mi>a</mi><mi>i</mi></msub><mo>⁢</mo><msub><mi>ρ</mi><mi>i</mi></msub></mrow><mo>-</mo><msub><mi>b</mi><mi>i</mi></msub></mrow></mrow></mrow><mo>,</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td> </tr> </table> <p class="ltx_p">denoting <math id="S0.SS1.p2.m4" class="ltx_Math" alttext="[\cdot]" display="inline"><mrow><mo stretchy="false">[</mo><mo>⋅</mo><mo stretchy="false">]</mo></mrow></math> the integral part <a class="nnexus_concept" href="http://planetmath.org/function">function</a>. Clearly <math id="S0.SS1.p2.m5" class="ltx_Math" alttext="\rho_{i+1}" display="inline"><msub><mi>ρ</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></math> is the <a class="nnexus_concept" href="http://planetmath.org/fractionalpart">fractional part</a> of <math id="S0.SS1.p2.m6" class="ltx_Math" alttext="a_{i}\rho_{i}" display="inline"><mrow><msub><mi>a</mi><mi>i</mi></msub><mo>⁢</mo><msub><mi>ρ</mi><mi>i</mi></msub></mrow></math>, therefore for all <math id="S0.SS1.p2.m7" class="ltx_Math" alttext="i\geq 1" display="inline"><mrow><mi>i</mi><mo>≥</mo><mn>1</mn></mrow></math> we have,</p> <table id="S0.E4" 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.E4.m1" class="ltx_Math" alttext="0\leq\rho_{i}<1." display="block"><mrow><mrow><mn>0</mn><mo>≤</mo><msub><mi>ρ</mi><mi>i</mi></msub><mo><</mo><mn>1</mn></mrow><mo>.</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td> </tr> </table> <p class="ltx_p">Next we multiply (4) by <math id="S0.SS1.p2.m8" class="ltx_Math" alttext="a_{i}" display="inline"><msub><mi>a</mi><mi>i</mi></msub></math> whence <math id="S0.SS1.p2.m9" class="ltx_Math" alttext="0\leq a_{i}\rho_{i}<a_{i}" display="inline"><mrow><mn>0</mn><mo>≤</mo><mrow><msub><mi>a</mi><mi>i</mi></msub><mo>⁢</mo><msub><mi>ρ</mi><mi>i</mi></msub></mrow><mo><</mo><msub><mi>a</mi><mi>i</mi></msub></mrow></math>, but <math id="S0.SS1.p2.m10" class="ltx_Math" alttext="b_{i}=[a_{i}\rho_{i}]" display="inline"><mrow><msub><mi>b</mi><mi>i</mi></msub><mo>=</mo><mrow><mo stretchy="false">[</mo><mrow><msub><mi>a</mi><mi>i</mi></msub><mo>⁢</mo><msub><mi>ρ</mi><mi>i</mi></msub></mrow><mo stretchy="false">]</mo></mrow></mrow></math>, so that</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="0\leq b_{i}\leq a_{i}-1," display="block"><mrow><mrow><mn>0</mn><mo>≤</mo><msub><mi>b</mi><mi>i</mi></msub><mo>≤</mo><mrow><msub><mi>a</mi><mi>i</mi></msub><mo>-</mo><mn>1</mn></mrow></mrow><mo>,</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </table> <p class="ltx_p">an inequality required by the theorem. <br class="ltx_break">Now, by (3) and (4), and applying <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">induction</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Induction.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/induction"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> on <math id="S0.SS1.p2.m11" class="ltx_Math" alttext="\rho_{i}" display="inline"><msub><mi>ρ</mi><mi>i</mi></msub></math>, we can establish that</p> <table id="S0.Ex2" 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.Ex2.m1" class="ltx_Math" alttext="\rho=b_{0}+\rho_{1}=b_{0}+\frac{b_{1}}{a_{1}}+\frac{\rho_{2}}{a_{1}}=b_{0}+% \frac{b_{1}}{a_{1}}+\frac{b_{2}}{a_{1}a_{2}}+\frac{\rho_{3}}{a_{1}a_{2}}" display="block"><mrow><mi>ρ</mi><mo>=</mo><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><msub><mi>ρ</mi><mn>1</mn></msub></mrow><mo>=</mo><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><mfrac><msub><mi>b</mi><mn>1</mn></msub><msub><mi>a</mi><mn>1</mn></msub></mfrac><mo>+</mo><mfrac><msub><mi>ρ</mi><mn>2</mn></msub><msub><mi>a</mi><mn>1</mn></msub></mfrac></mrow><mo>=</mo><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><mfrac><msub><mi>b</mi><mn>1</mn></msub><msub><mi>a</mi><mn>1</mn></msub></mfrac><mo>+</mo><mfrac><msub><mi>b</mi><mn>2</mn></msub><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>⁢</mo><msub><mi>a</mi><mn>2</mn></msub></mrow></mfrac><mo>+</mo><mfrac><msub><mi>ρ</mi><mn>3</mn></msub><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>⁢</mo><msub><mi>a</mi><mn>2</mn></msub></mrow></mfrac></mrow></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </table> <table id="S0.E5" 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.E5.m1" class="ltx_Math" alttext="=\cdots=b_{0}+\sum_{i=1}^{n}\frac{b_{i}}{\prod_{j=1}^{i}a_{j}}+\frac{\rho_{n+1% }}{\prod_{j=1}^{n}a_{j}}." display="block"><mrow><mrow><mi></mi><mo>=</mo><mi mathvariant="normal">⋯</mi><mo>=</mo><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfrac><msub><mi>b</mi><mi>i</mi></msub><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mi>j</mi></msub></mrow></mfrac></mrow><mo>+</mo><mfrac><msub><mi>ρ</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>a</mi><mi>j</mi></msub></mrow></mfrac></mrow></mrow><mo>.</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td> </tr> </table> <p class="ltx_p">Now we define</p> <table id="S0.Ex3" 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.Ex3.m1" class="ltx_Math" alttext="d_{n}=b_{0}+\sum_{i=1}^{n}\frac{b_{i}}{\prod_{j=1}^{i}a_{j}}," display="block"><mrow><mrow><msub><mi>d</mi><mi>n</mi></msub><mo>=</mo><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfrac><msub><mi>b</mi><mi>i</mi></msub><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mi>j</mi></msub></mrow></mfrac></mrow></mrow></mrow><mo>,</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </table> <p class="ltx_p">thus from (4), (5) and by the <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">hypothesis</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Hypothesis.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/deduction"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> <math id="S0.SS1.p2.m12" class="ltx_Math" alttext="a_{i}\geq 2" display="inline"><mrow><msub><mi>a</mi><mi>i</mi></msub><mo>≥</mo><mn>2</mn></mrow></math>, we arrive to</p> <table id="S0.Ex4" 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.Ex4.m1" class="ltx_Math" alttext="0\leq\rho-d_{n}=\frac{\rho_{n+1}}{\prod_{j=1}^{n}a_{j}}<\frac{1}{2^{n}}," display="block"><mrow><mrow><mn>0</mn><mo>≤</mo><mrow><mi>ρ</mi><mo>-</mo><msub><mi>d</mi><mi>n</mi></msub></mrow><mo>=</mo><mfrac><msub><mi>ρ</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>a</mi><mi>j</mi></msub></mrow></mfrac><mo><</mo><mfrac><mn>1</mn><msup><mn>2</mn><mi>n</mi></msup></mfrac></mrow><mo>,</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </table> <p class="ltx_p">because the fractional part <math id="S0.SS1.p2.m13" class="ltx_Math" alttext="\rho_{n+1}<1" display="inline"><mrow><msub><mi>ρ</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo><</mo><mn>1</mn></mrow></math>. Then if we let <math id="S0.SS1.p2.m14" class="ltx_Math" alttext="n" display="inline"><mi>n</mi></math> grows beyond of any bound, <math id="S0.SS1.p2.m15" class="ltx_Math" alttext="\rho-d_{n}" display="inline"><mrow><mi>ρ</mi><mo>-</mo><msub><mi>d</mi><mi>n</mi></msub></mrow></math> will be so close to zero as we want and such a observation implies the representation (1). <br class="ltx_break">We still need to prove the another inequality of the theorem, i.e. <math id="S0.SS1.p2.m16" class="ltx_Math" alttext="b_{i}<a_{i}-1" display="inline"><mrow><msub><mi>b</mi><mi>i</mi></msub><mo><</mo><mrow><msub><mi>a</mi><mi>i</mi></msub><mo>-</mo><mn>1</mn></mrow></mrow></math> for infinitely many <math id="S0.SS1.p2.m17" class="ltx_Math" alttext="i" display="inline"><mi>i</mi></math>, but also the uniqueness of representation (1). To do that we need to make use of the <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">identity</a><sup style="display: none;"><a class="nnexus_concept" href="http://planetmath.org/equality"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/multivaluedfunction"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup></p> <table id="S0.E6" 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.E6.m1" class="ltx_Math" alttext="\sum_{i=1}^{\infty}\frac{a_{n+i}-1}{\prod_{j=1}^{i}a_{n+j}}=1." display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mrow><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub><mo>-</mo><mn>1</mn></mrow><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>j</mi></mrow></msub></mrow></mfrac></mrow><mo>=</mo><mn>1</mn></mrow><mo>.</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6)</span></td> </tr> </table> <p class="ltx_p">(It is legitimate to consider this identity as a lemma, as we need it to prove this theorem as well as the next one). <br class="ltx_break">We shall prove later this identity. Let us prove the inequality <math id="S0.SS1.p2.m18" class="ltx_Math" alttext="b_{i}<a_{i}-1" display="inline"><mrow><msub><mi>b</mi><mi>i</mi></msub><mo><</mo><mrow><msub><mi>a</mi><mi>i</mi></msub><mo>-</mo><mn>1</mn></mrow></mrow></math> by <em class="ltx_emph ltx_font_italic">tertio excluso</em>; thus suppose that there is a fixed integer <math id="S0.SS1.p2.m19" class="ltx_Math" alttext="n" display="inline"><mi>n</mi></math> such that <math id="S0.SS1.p2.m20" class="ltx_Math" alttext="b_{i}=a_{i}-1" display="inline"><mrow><msub><mi>b</mi><mi>i</mi></msub><mo>=</mo><mrow><msub><mi>a</mi><mi>i</mi></msub><mo>-</mo><mn>1</mn></mrow></mrow></math> for all <math id="S0.SS1.p2.m21" class="ltx_Math" alttext="i>n" display="inline"><mrow><mi>i</mi><mo>></mo><mi>n</mi></mrow></math>. From (1) and (6) we get</p> <table id="A0.EGx1" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table"> <tr id="S0.Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="S0.Ex5.m1" class="ltx_Math" alttext="\displaystyle\rho=" display="inline"><mrow><mi>ρ</mi><mo>=</mo><mi></mi></mrow></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="S0.Ex5.m2" class="ltx_Math" alttext="\displaystyle b_{0}+\sum_{i=1}^{n}\frac{b_{i}}{\prod_{j=1}^{i}a_{j}}+\sum_{i=n% +1}^{\infty}\frac{a_{i}-1}{\prod_{j=1}^{i}a_{j}}" display="inline"><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover></mstyle><mstyle displaystyle="true"><mfrac><msub><mi>b</mi><mi>i</mi></msub><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mi>j</mi></msub></mrow></mfrac></mstyle></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mstyle displaystyle="true"><mfrac><mrow><msub><mi>a</mi><mi>i</mi></msub><mo>-</mo><mn>1</mn></mrow><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mi>j</mi></msub></mrow></mfrac></mstyle></mrow></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr id="S0.Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="S0.Ex6.m1" class="ltx_Math" alttext="\displaystyle=" display="inline"><mo>=</mo></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="S0.Ex6.m2" class="ltx_Math" alttext="\displaystyle b_{0}+\sum_{i=1}^{n}\frac{b_{i}}{\prod_{j=1}^{i}a_{j}}+\frac{1}{% \prod_{j=1}^{n}a_{j}}\sum_{i=1}^{\infty}\frac{a_{n+i}-1}{\prod_{j=1}^{i}a_{n+j}}" display="inline"><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover></mstyle><mstyle displaystyle="true"><mfrac><msub><mi>b</mi><mi>i</mi></msub><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mi>j</mi></msub></mrow></mfrac></mstyle></mrow><mo>+</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>a</mi><mi>j</mi></msub></mrow></mfrac></mstyle><mo>⁢</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover></mstyle><mstyle displaystyle="true"><mfrac><mrow><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub><mo>-</mo><mn>1</mn></mrow><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>j</mi></mrow></msub></mrow></mfrac></mstyle></mrow></mrow></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr id="S0.Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="S0.Ex7.m1" class="ltx_Math" alttext="\displaystyle=" display="inline"><mo>=</mo></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="S0.Ex7.m2" class="ltx_Math" alttext="\displaystyle b_{0}+\sum_{i=1}^{n}\frac{b_{i}}{\prod_{j=1}^{i}a_{j}}+\frac{1}{% \prod_{j=1}^{n}a_{j}}," display="inline"><mrow><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><mrow><mstyle displaystyle="true"><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover></mstyle><mstyle displaystyle="true"><mfrac><msub><mi>b</mi><mi>i</mi></msub><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mi>j</mi></msub></mrow></mfrac></mstyle></mrow><mo>+</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>a</mi><mi>j</mi></msub></mrow></mfrac></mstyle></mrow><mo>,</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </table> <p class="ltx_p">and comparing this with (5) one realizes that <math id="S0.SS1.p2.m22" class="ltx_Math" alttext="\rho_{n+1}=1" display="inline"><mrow><msub><mi>ρ</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn></mrow></math>, contradicting (4). <br class="ltx_break">Finally we must prove the uniqueness of the representation (1). So that we suppose</p> <table id="S0.Ex8" 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.Ex8.m1" class="ltx_Math" alttext="\rho=c_{0}+\sum_{i=1}^{\infty}\frac{c_{i}}{\prod_{j=1}^{i}a_{j}}." display="block"><mrow><mrow><mi>ρ</mi><mo>=</mo><mrow><msub><mi>c</mi><mn>0</mn></msub><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msub><mi>c</mi><mi>i</mi></msub><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mi>j</mi></msub></mrow></mfrac></mrow></mrow></mrow><mo>.</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </table> <p class="ltx_p">Here the integers <math id="S0.SS1.p2.m23" class="ltx_Math" alttext="c_{i}" display="inline"><msub><mi>c</mi><mi>i</mi></msub></math> <a class="nnexus_concept" href="http://planetmath.org/satisfactionrelation">satisfy</a> the same conditions as do the <math id="S0.SS1.p2.m24" class="ltx_Math" alttext="b_{i}" display="inline"><msub><mi>b</mi><mi>i</mi></msub></math>. It is necessary (and <a class="nnexus_concept" href="http://planetmath.org/necessaryandsufficient">sufficient</a>!) to show that <math id="S0.SS1.p2.m25" class="ltx_Math" alttext="c_{i}=b_{i}" display="inline"><mrow><msub><mi>c</mi><mi>i</mi></msub><mo>=</mo><msub><mi>b</mi><mi>i</mi></msub></mrow></math> for every <math id="S0.SS1.p2.m26" class="ltx_Math" alttext="i" display="inline"><mi>i</mi></math>. The condition <math id="S0.SS1.p2.m27" class="ltx_Math" alttext="c_{i}<a_{i}-1" display="inline"><mrow><msub><mi>c</mi><mi>i</mi></msub><mo><</mo><mrow><msub><mi>a</mi><mi>i</mi></msub><mo>-</mo><mn>1</mn></mrow></mrow></math>, for infinitely many <math id="S0.SS1.p2.m28" class="ltx_Math" alttext="i" display="inline"><mi>i</mi></math>, altogether with the identity (6) imply that</p> <table id="S0.Ex9" 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.Ex9.m1" class="ltx_Math" alttext="\sum_{i=1}^{\infty}\frac{c_{i}}{\prod_{j=1}^{i}a_{j}}<1," display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msub><mi>c</mi><mi>i</mi></msub><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mi>j</mi></msub></mrow></mfrac></mrow><mo><</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </table> <p class="ltx_p">so we see that <math id="S0.SS1.p2.m29" class="ltx_Math" alttext="c_{0}" display="inline"><msub><mi>c</mi><mn>0</mn></msub></math> is the integral part of <math id="S0.SS1.p2.m30" class="ltx_Math" alttext="\rho" display="inline"><mi>ρ</mi></math>, i.e. <math id="S0.SS1.p2.m31" class="ltx_Math" alttext="c_{0}=[\rho]" display="inline"><mrow><msub><mi>c</mi><mn>0</mn></msub><mo>=</mo><mrow><mo stretchy="false">[</mo><mi>ρ</mi><mo stretchy="false">]</mo></mrow></mrow></math>, but also from (2) <math id="S0.SS1.p2.m32" class="ltx_Math" alttext="b_{0}=[\rho]" display="inline"><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>=</mo><mrow><mo stretchy="false">[</mo><mi>ρ</mi><mo stretchy="false">]</mo></mrow></mrow></math>, therefore <math id="S0.SS1.p2.m33" class="ltx_Math" alttext="c_{0}=b_{0}" display="inline"><mrow><msub><mi>c</mi><mn>0</mn></msub><mo>=</mo><msub><mi>b</mi><mn>0</mn></msub></mrow></math>. Next we shall again use <em class="ltx_emph ltx_font_italic">tertio excluso</em>. On this way let us suppose that for some <math id="S0.SS1.p2.m34" class="ltx_Math" alttext="n\geq 1" display="inline"><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math> the pair <math id="S0.SS1.p2.m35" class="ltx_Math" alttext="b_{n}" display="inline"><msub><mi>b</mi><mi>n</mi></msub></math> and <math id="S0.SS1.p2.m36" class="ltx_Math" alttext="c_{n}" display="inline"><msub><mi>c</mi><mi>n</mi></msub></math> are unequal. There is no loss of generality in assuming that <math id="S0.SS1.p2.m37" class="ltx_Math" alttext="n" display="inline"><mi>n</mi></math> is the smallest integer with this property (which is justified by a simple inductive <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">argument</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Argument.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/argument"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup>), and that <math id="S0.SS1.p2.m38" class="ltx_Math" alttext="b_{n}>c_{n}" display="inline"><mrow><msub><mi>b</mi><mi>n</mi></msub><mo>></mo><msub><mi>c</mi><mi>n</mi></msub></mrow></math>, so that <math id="S0.SS1.p2.m39" class="ltx_Math" alttext="b_{n}-c_{n}\geq 1" display="inline"><mrow><mrow><msub><mi>b</mi><mi>n</mi></msub><mo>-</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>≥</mo><mn>1</mn></mrow></math>. Thus we have</p> <table id="S0.Ex10" 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.Ex10.m1" class="ltx_Math" alttext="\sum_{i=n}^{\infty}\frac{b_{i}}{\prod_{j=1}^{i}a_{j}}=\sum_{i=n}^{\infty}\frac% {c_{i}}{\prod_{j=1}^{i}a_{j}}." display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mi>n</mi></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msub><mi>b</mi><mi>i</mi></msub><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mi>j</mi></msub></mrow></mfrac></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mi>n</mi></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msub><mi>c</mi><mi>i</mi></msub><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mi>j</mi></msub></mrow></mfrac></mrow></mrow><mo>.</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </table> <p class="ltx_p">(There is no <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">contradiction</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Contradiction.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/contradiction"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/contradictorystatement"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> at all in this equality, as it is easily seen in the next below equation). <br class="ltx_break">It is obvious that these series are <a class="nnexus_concept" href="http://planetmath.org/convergentseries">absolutely convergent</a>, so we may rearrange terms to obtain</p> <table id="S0.Ex11" 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.Ex11.m1" class="ltx_Math" alttext="\sum_{i=n+1}^{\infty}\frac{c_{i}-b_{i}}{\prod_{j=1}^{i}a_{j}}=\frac{b_{n}-c_{n% }}{\prod_{j=1}^{n}a_{j}}\geq\frac{1}{\prod_{j=1}^{n}a_{j}}." display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mrow><msub><mi>c</mi><mi>i</mi></msub><mo>-</mo><msub><mi>b</mi><mi>i</mi></msub></mrow><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mi>j</mi></msub></mrow></mfrac></mrow><mo>=</mo><mfrac><mrow><msub><mi>b</mi><mi>n</mi></msub><mo>-</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>a</mi><mi>j</mi></msub></mrow></mfrac><mo>≥</mo><mfrac><mn>1</mn><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>a</mi><mi>j</mi></msub></mrow></mfrac></mrow><mo>.</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </table> <p class="ltx_p">But then recalling that <math id="S0.SS1.p2.m40" class="ltx_Math" alttext="c_{i}<a_{i}-1" display="inline"><mrow><msub><mi>c</mi><mi>i</mi></msub><mo><</mo><mrow><msub><mi>a</mi><mi>i</mi></msub><mo>-</mo><mn>1</mn></mrow></mrow></math>, we see tat <math id="S0.SS1.p2.m41" class="ltx_Math" alttext="c_{i}-b_{i}<a_{i}-1" display="inline"><mrow><mrow><msub><mi>c</mi><mi>i</mi></msub><mo>-</mo><msub><mi>b</mi><mi>i</mi></msub></mrow><mo><</mo><mrow><msub><mi>a</mi><mi>i</mi></msub><mo>-</mo><mn>1</mn></mrow></mrow></math>. From this fact and (6), we can write</p> <table id="S0.Ex12" 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.Ex12.m1" class="ltx_Math" alttext="\sum_{i=n+1}^{\infty}\frac{c_{i}-b_{i}}{\prod_{j=1}^{i}a_{j}}<\sum_{i=n+1}^{% \infty}\frac{a_{i}-1}{\prod_{j=1}^{i}a_{j}}" display="block"><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mrow><msub><mi>c</mi><mi>i</mi></msub><mo>-</mo><msub><mi>b</mi><mi>i</mi></msub></mrow><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mi>j</mi></msub></mrow></mfrac></mrow><mo><</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mrow><msub><mi>a</mi><mi>i</mi></msub><mo>-</mo><mn>1</mn></mrow><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mi>j</mi></msub></mrow></mfrac></mrow></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </table> <table id="S0.Ex13" 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.Ex13.m1" class="ltx_Math" alttext="=\frac{1}{\prod_{j=1}^{n}a_{j}}\sum_{i=1}^{\infty}\frac{a_{n+i}-1}{\prod_{j=1}% ^{i}a_{n+j}}=\frac{1}{\prod_{j=1}^{n}a_{j}}." display="block"><mrow><mrow><mi></mi><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>a</mi><mi>j</mi></msub></mrow></mfrac><mo>⁢</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mrow><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub><mo>-</mo><mn>1</mn></mrow><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>j</mi></mrow></msub></mrow></mfrac></mrow></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>a</mi><mi>j</mi></msub></mrow></mfrac></mrow><mo>.</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </table> <p class="ltx_p">This is a contradiction with respect to the inequality found before, thus the proof of this theorem is <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">complete</a><sup style="display: none;"><a class="nnexus_concept" href="http://planetmath.org/ordersinanumberfield"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/soundcomplete"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/kripkesemantics"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/maximallyconsistent"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup>. ∎</p> </div> </div> </section> <section id="S0.SS2" class="ltx_subsection"> <h2 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">0.2 </span>Some implications</h2> <div id="S0.SS2.p1" class="ltx_para"> <p class="ltx_p">First we remarked that Theorem 1 is a <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">generalization</a><sup style="display: none;"><a class="nnexus_concept" href="http://planetmath.org/hilbertsystem"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/axiomsystemforfirstorderlogic"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> of the standard <a class="nnexus_concept" href="http://planetmath.org/decimalexpansion">decimal expansion</a> for a real number <math id="S0.SS2.p1.m1" class="ltx_Math" alttext="\rho" display="inline"><mi>ρ</mi></math>. This may be seen by taking all the integers <math id="S0.SS2.p1.m2" class="ltx_Math" alttext="a_{i}=10" display="inline"><mrow><msub><mi>a</mi><mi>i</mi></msub><mo>=</mo><mn>10</mn></mrow></math>. Thus, if <math id="S0.SS2.p1.m3" class="ltx_Math" alttext="\rho>0" display="inline"><mrow><mi>ρ</mi><mo>></mo><mn>0</mn></mrow></math>, (1) gives the decimal representation</p> <table id="S0.E7" 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.E7.m1" class="ltx_Math" alttext="\rho=b_{0}+\sum_{i=1}^{\infty}\frac{b_{i}}{10^{i}}=b_{0}.b_{1}b_{2}\cdots." display="block"><mrow><mrow><mrow><mi>ρ</mi><mo>=</mo><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msub><mi>b</mi><mi>i</mi></msub><msup><mn>10</mn><mi>i</mi></msup></mfrac></mrow></mrow><mo>=</mo><msub><mi>b</mi><mn>0</mn></msub></mrow><mo>.</mo><mrow><msub><mi>b</mi><mn>1</mn></msub><mo>⁢</mo><msub><mi>b</mi><mn>2</mn></msub><mo>⁢</mo><mi mathvariant="normal">⋯</mi></mrow></mrow><mo>.</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(7)</span></td> </tr> </table> <p class="ltx_p">Second, if <math id="S0.SS2.p1.m4" class="ltx_Math" alttext="\rho<0" display="inline"><mrow><mi>ρ</mi><mo><</mo><mn>0</mn></mrow></math>, we must write its decimal representation on the form (7)and then changing all signs. Third, any real <math id="S0.SS2.p1.m5" class="ltx_Math" alttext="\rho" display="inline"><mi>ρ</mi></math> could have an ambiguous decimal representation, e.g. <math id="S0.SS2.p1.m6" class="ltx_Math" alttext="\rho=1.5699\cdots" display="inline"><mrow><mi>ρ</mi><mo>=</mo><mrow><mn>1.5699</mn><mo>⁢</mo><mi mathvariant="normal">⋯</mi></mrow></mrow></math>, having an <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">infinite</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Infinite.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/infinite"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> successive sequence of <math id="S0.SS2.p1.m7" class="ltx_Math" alttext="9" display="inline"><mn>9</mn></math>’s, which also involves a <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">geometric series</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/GeometricSeries.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/geometricseries"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> in <math id="S0.SS2.p1.m8" class="ltx_Math" alttext="10^{-i}" display="inline"><msup><mn>10</mn><mrow><mo>-</mo><mi>i</mi></mrow></msup></math> in turns implying (at the limit) that also <math id="S0.SS2.p1.m9" class="ltx_Math" alttext="\rho=1.57" display="inline"><mrow><mi>ρ</mi><mo>=</mo><mn>1.57</mn></mrow></math>. For that reason, (7) represents that number with an infinite succession of <math id="S0.SS2.p1.m10" class="ltx_Math" alttext="0" display="inline"><mn>0</mn></math>,s, that is, <math id="S0.SS2.p1.m11" class="ltx_Math" alttext="\rho=1.57=1.5700\cdots" display="inline"><mrow><mi>ρ</mi><mo>=</mo><mn>1.57</mn><mo>=</mo><mrow><mn>1.5700</mn><mo>⁢</mo><mi mathvariant="normal">⋯</mi></mrow></mrow></math>. The reason for this resides in that an infinite succession of <math id="S0.SS2.p1.m12" class="ltx_Math" alttext="9" display="inline"><mn>9</mn></math>’s is ruled out by the condition of the theorem that <math id="S0.SS2.p1.m13" class="ltx_Math" alttext="b_{i}<a_{i}-1" display="inline"><mrow><msub><mi>b</mi><mi>i</mi></msub><mo><</mo><mrow><msub><mi>a</mi><mi>i</mi></msub><mo>-</mo><mn>1</mn></mrow></mrow></math> for infinitely many <math id="S0.SS2.p1.m14" class="ltx_Math" alttext="i" display="inline"><mi>i</mi></math>, a condition that in the present example takes the form <math id="S0.SS2.p1.m15" class="ltx_Math" alttext="b_{i}<9" display="inline"><mrow><msub><mi>b</mi><mi>i</mi></msub><mo><</mo><mn>9</mn></mrow></math> for infinitely many <math id="S0.SS2.p1.m16" class="ltx_Math" alttext="i" display="inline"><mi>i</mi></math>. <br class="ltx_break">Now we prove (6).</p> </div> <div id="Thmlemma1" class="ltx_theorem ltx_theorem_lemma"> <h6 class="ltx_title ltx_runin ltx_font_bold ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem">Lemma 1</span>.</h6> <div id="Thmlemma1.p1" class="ltx_para"> <p class="ltx_p"><span class="ltx_text ltx_font_italic">For the <a class="nnexus_concept" href="http://mathworld.wolfram.com/IntegerSequence.html">integers sequence</a> <math id="Thmlemma1.p1.m1" class="ltx_Math" alttext="\{a_{i}\}" display="inline"><mrow><mo mathvariant="normal" stretchy="false">{</mo><msub><mi>a</mi><mi>i</mi></msub><mo mathvariant="normal" stretchy="false">}</mo></mrow></math>, where <math id="Thmlemma1.p1.m2" class="ltx_Math" alttext="a_{i}>1" display="inline"><mrow><msub><mi>a</mi><mi>i</mi></msub><mo mathvariant="normal">></mo><mn mathvariant="normal">1</mn></mrow></math> for every <math id="Thmlemma1.p1.m3" class="ltx_Math" alttext="i\geq 1" display="inline"><mrow><mi>i</mi><mo mathvariant="normal">≥</mo><mn mathvariant="normal">1</mn></mrow></math>, we have</span></p> <table id="S0.Ex14" 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.Ex14.m1" class="ltx_Math" alttext="\sum_{i=1}^{\infty}\frac{a_{n+i}-1}{\prod_{j=1}^{i}a_{n+j}}=1." display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mrow><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub><mo>-</mo><mn>1</mn></mrow><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>j</mi></mrow></msub></mrow></mfrac></mrow><mo>=</mo><mn>1</mn></mrow><mo>.</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </table> </div> </div> <div class="ltx_proof"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div id="S0.SS2.p2" class="ltx_para"> <p class="ltx_p">Let us take the <a class="nnexus_concept" href="http://planetmath.org/sumofseries">partial sum</a></p> <table id="S0.Ex15" 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.Ex15.m1" class="ltx_Math" alttext="S_{m}=\sum_{i=1}^{m}\frac{a_{n+i}-1}{\prod_{j=1}^{i}a_{n+j}}=\sum_{i=1}^{m}% \frac{a_{n+i}}{\prod_{j=1}^{i}a_{n+j}}-\sum_{i=1}^{m}\frac{1}{\prod_{j=1}^{i}a% _{n+j}}" display="block"><mrow><msub><mi>S</mi><mi>m</mi></msub><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><mfrac><mrow><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub><mo>-</mo><mn>1</mn></mrow><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>j</mi></mrow></msub></mrow></mfrac></mrow><mo>=</mo><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><mfrac><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>j</mi></mrow></msub></mrow></mfrac></mrow><mo>-</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><mfrac><mn>1</mn><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>j</mi></mrow></msub></mrow></mfrac></mrow></mrow></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </table> <table id="S0.Ex16" 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.Ex16.m1" class="ltx_Math" alttext="=\frac{a_{n+1}}{a_{n+1}}+\sum_{i=2}^{m}\frac{a_{n+i}}{\prod_{j=1}^{i}a_{n+j}}-% \sum_{i=1}^{m}\frac{1}{\prod_{j=1}^{i}a_{n+j}}" display="block"><mrow><mi></mi><mo>=</mo><mrow><mrow><mfrac><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mfrac><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>2</mn></mrow><mi>m</mi></munderover><mfrac><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>j</mi></mrow></msub></mrow></mfrac></mrow></mrow><mo>-</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><mfrac><mn>1</mn><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>j</mi></mrow></msub></mrow></mfrac></mrow></mrow></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </table> <table id="S0.Ex17" 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.Ex17.m1" class="ltx_Math" alttext="=1+\sum_{i=2}^{m}\frac{a_{n+i}}{\prod_{j=1}^{i}a_{n+j}}-\sum_{i=2}^{m+1}\frac{% 1}{\prod_{j=1}^{i}a_{n+j-1}}" display="block"><mrow><mi></mi><mo>=</mo><mrow><mrow><mn>1</mn><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>2</mn></mrow><mi>m</mi></munderover><mfrac><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>j</mi></mrow></msub></mrow></mfrac></mrow></mrow><mo>-</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>2</mn></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></munderover><mfrac><mn>1</mn><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mrow><mrow><mi>n</mi><mo>+</mo><mi>j</mi></mrow><mo>-</mo><mn>1</mn></mrow></msub></mrow></mfrac></mrow></mrow></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </table> <table id="S0.Ex18" 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.Ex18.m1" class="ltx_Math" alttext="=1+\sum_{i=2}^{m}\frac{a_{n+i}}{\prod_{j=1}^{i}a_{n+j}}-\sum_{i=2}^{m+1}\frac{% a_{n+i}}{a_{n+i}\prod_{j=1}^{i}a_{n+j-1}}" display="block"><mrow><mi></mi><mo>=</mo><mrow><mrow><mn>1</mn><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>2</mn></mrow><mi>m</mi></munderover><mfrac><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>j</mi></mrow></msub></mrow></mfrac></mrow></mrow><mo>-</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>2</mn></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></munderover><mfrac><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub><mrow><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub><mo>⁢</mo><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mrow><mrow><mi>n</mi><mo>+</mo><mi>j</mi></mrow><mo>-</mo><mn>1</mn></mrow></msub></mrow></mrow></mfrac></mrow></mrow></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </table> <table id="S0.Ex19" 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.Ex19.m1" class="ltx_Math" alttext="=1+\sum_{i=2}^{m}\frac{a_{n+i}}{\prod_{j=1}^{i}a_{n+j}}-\sum_{i=2}^{m}\frac{a_% {n+i}}{\prod_{j=1}^{i}a_{n+j}}-\frac{n+m+1}{(n+m+1)\prod_{j=1}^{m+1}a_{n+j-1}}" display="block"><mrow><mi></mi><mo>=</mo><mrow><mrow><mn>1</mn><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>2</mn></mrow><mi>m</mi></munderover><mfrac><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>j</mi></mrow></msub></mrow></mfrac></mrow></mrow><mo>-</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>2</mn></mrow><mi>m</mi></munderover><mfrac><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>j</mi></mrow></msub></mrow></mfrac></mrow><mo>-</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mi>m</mi><mo>+</mo><mn>1</mn></mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>n</mi><mo>+</mo><mi>m</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mo>⁢</mo><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msubsup><msub><mi>a</mi><mrow><mrow><mi>n</mi><mo>+</mo><mi>j</mi></mrow><mo>-</mo><mn>1</mn></mrow></msub></mrow></mrow></mfrac></mrow></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </table> <table id="S0.Ex20" 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.Ex20.m1" class="ltx_Math" alttext="=1+\sum_{i=2}^{m}\frac{a_{n+i}}{\prod_{j=1}^{i}a_{n+j}}-\sum_{i=2}^{m}\frac{a_% {n+i}}{\prod_{j=1}^{i}a_{n+j}}-\frac{1}{\prod_{j=1}^{m}a_{n+j}}," display="block"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mn>1</mn><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>2</mn></mrow><mi>m</mi></munderover><mfrac><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>j</mi></mrow></msub></mrow></mfrac></mrow></mrow><mo>-</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>2</mn></mrow><mi>m</mi></munderover><mfrac><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>j</mi></mrow></msub></mrow></mfrac></mrow><mo>-</mo><mfrac><mn>1</mn><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>j</mi></mrow></msub></mrow></mfrac></mrow></mrow><mo>,</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </table> <p class="ltx_p">since when in the <math id="S0.SS2.p2.m1" class="ltx_Math" alttext="\prod" display="inline"><mo largeop="true" symmetric="true">∏</mo></math> <a class="nnexus_concept" href="http://planetmath.org/operator">operator</a> <math id="S0.SS2.p2.m2" class="ltx_Math" alttext="a_{n+j-1}\mapsto a_{n+j}" display="inline"><mrow><msub><mi>a</mi><mrow><mrow><mi>n</mi><mo>+</mo><mi>j</mi></mrow><mo>-</mo><mn>1</mn></mrow></msub><mo>↦</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>j</mi></mrow></msub></mrow></math>, then the index value of <math id="S0.SS2.p2.m3" class="ltx_Math" alttext="j" display="inline"><mi>j</mi></math>, at its <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">upper limit</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/UpperLimit.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/definiteintegral"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup>, <math id="S0.SS2.p2.m4" class="ltx_Math" alttext="m+1\mapsto m" display="inline"><mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow><mo>↦</mo><mi>m</mi></mrow></math>, but its <a class="nnexus_concept" href="http://mathworld.wolfram.com/LowerLimit.html">lower limit</a> does not. Thus, central sums cancel and the last term <a class="nnexus_concept" href="http://planetmath.org/zeroofafunction">vanishes</a> because, by hypothesis, we have</p> <table id="S0.Ex21" 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.Ex21.m1" class="ltx_Math" alttext="\lim_{m\to\infty}S_{m}=\sum_{i=1}^{\infty}\frac{a_{n+i}-1}{\prod_{j=1}^{i}a_{n% +j}}=1-\lim_{m\to\infty}\frac{1}{\prod_{j=1}^{m}a_{n+j}}," display="block"><mrow><mrow><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>m</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo>⁡</mo><msub><mi>S</mi><mi>m</mi></msub></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mrow><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub><mo>-</mo><mn>1</mn></mrow><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>j</mi></mrow></msub></mrow></mfrac></mrow><mo>=</mo><mrow><mn>1</mn><mo>-</mo><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>m</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo>⁡</mo><mfrac><mn>1</mn><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>j</mi></mrow></msub></mrow></mfrac></mrow></mrow></mrow><mo>,</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </table> <p class="ltx_p">and the lemma is proved. ∎</p> </div> </div> <div id="S0.SS2.p3" class="ltx_para"> <p class="ltx_p">Theorem 1 also represents an <em class="ltx_emph ltx_font_italic"><a class="nnexus_concept" href="http://planetmath.org/irrational">irrational</a></em> number whenever we add a couple of additional conditions. Thus we have the following important theorem.</p> </div> <div id="Thmtheorem2" class="ltx_theorem ltx_theorem_theorem"> <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="Thmtheorem2.p1" class="ltx_para"> <p class="ltx_p"><span class="ltx_text ltx_font_italic">Let us consider the same integers sequence <math id="Thmtheorem2.p1.m1" class="ltx_Math" alttext="\{a_{i}\}" display="inline"><mrow><mo mathvariant="normal" stretchy="false">{</mo><msub><mi>a</mi><mi>i</mi></msub><mo mathvariant="normal" stretchy="false">}</mo></mrow></math> described in the preceding theorem, and that the integers <math id="Thmtheorem2.p1.m2" class="ltx_Math" alttext="b_{i}" display="inline"><msub><mi>b</mi><mi>i</mi></msub></math> satisfying the inequalities of that result. In <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">addition</a><sup style="display: none;"><a class="nnexus_concept" href="http://planetmath.org/addition"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/cardinalarithmetic"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup>, let us assume that infinite integers <math id="Thmtheorem2.p1.m3" class="ltx_Math" alttext="b_{i}" display="inline"><msub><mi>b</mi><mi>i</mi></msub></math> are <a class="nnexus_concept" href="http://planetmath.org/positive">positive</a>, and that each <em class="ltx_emph ltx_font_upright"><a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">prime number</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/PrimeNumber.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/prime"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup></em> divides infinitely many <math id="Thmtheorem2.p1.m4" class="ltx_Math" alttext="a_{i}" display="inline"><msub><mi>a</mi><mi>i</mi></msub></math>. Then <math id="Thmtheorem2.p1.m5" class="ltx_Math" alttext="\rho" display="inline"><mi>ρ</mi></math> is irrational.</span></p> </div> </div> <div class="ltx_proof"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div id="S0.SS2.p4" class="ltx_para"> <p class="ltx_p">We contradict the thesis by supposing <math id="S0.SS2.p4.m1" class="ltx_Math" alttext="\rho=p/q" display="inline"><mrow><mi>ρ</mi><mo>=</mo><mrow><mi>p</mi><mo>/</mo><mi>q</mi></mrow></mrow></math> is <a class="nnexus_concept" href="http://planetmath.org/rationalnumber">rational</a> (<math id="S0.SS2.p4.m2" class="ltx_Math" alttext="p" display="inline"><mi>p</mi></math>, <math id="S0.SS2.p4.m3" class="ltx_Math" alttext="q" display="inline"><mi>q</mi></math>, <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">coprime</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Coprime.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/coprime"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup>). By the last hypothesis in the preceding theorem, we can choose an integer <math id="S0.SS2.p4.m4" class="ltx_Math" alttext="n" display="inline"><mi>n</mi></math> sufficiently large in order to <math id="S0.SS2.p4.m5" class="ltx_Math" alttext="q" display="inline"><mi>q</mi></math> be a <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">divisor</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Divisor.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/divisibility"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/divisortheory"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> of <math id="S0.SS2.p4.m6" class="ltx_Math" alttext="\prod_{j=1}^{n}a_{j}" display="inline"><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>a</mi><mi>j</mi></msub></mrow></math>. Now we may use (1) replacing the LHS by our <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">rational number</a><sup style="display: none;"><a class="nnexus_concept" href="http://planetmath.org/egyptianfraction"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/liberabaci"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">assumption</a><sup style="display: none;"><a class="nnexus_concept" href="http://planetmath.org/derivationsinnaturaldeduction"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/deduction"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup>, next multiplying both side by the latter <a class="nnexus_concept" href="http://planetmath.org/product">product</a>, and rearranging terms we get (we do the <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">partition</a><sup style="display: none;"><a class="nnexus_concept" href="http://planetmath.org/partition1"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/integerpartition"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/partition"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> sum <math id="S0.SS2.p4.m7" class="ltx_Math" alttext="i=1,\ldots,n;n+1,\ldots\infty" display="inline"><mrow><mi>i</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mi>n</mi><mo>;</mo><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mo>,</mo><mrow><mi mathvariant="normal">…</mi><mo>⁢</mo><mi mathvariant="normal">∞</mi></mrow></mrow></mrow></math>)</p> <table id="S0.E8" 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.E8.m1" class="ltx_Math" alttext="\frac{\prod_{j=1}^{n}a_{j}(p-b_{0}q)}{q}-\sum_{i=1}^{n}\frac{b_{0}\prod_{j=1}^% {n}a_{j}}{\prod_{j=1}^{i}a_{j}}=\sum_{i=1}^{\infty}\frac{b_{n+i}}{\prod_{j=1}^% {i}a_{n+j}}." display="block"><mrow><mrow><mrow><mfrac><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mrow><msub><mi>a</mi><mi>j</mi></msub><mo>⁢</mo><mrow><mo stretchy="false">(</mo><mrow><mi>p</mi><mo>-</mo><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>⁢</mo><mi>q</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mi>q</mi></mfrac><mo>-</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfrac><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>⁢</mo><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>a</mi><mi>j</mi></msub></mrow></mrow><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mi>j</mi></msub></mrow></mfrac></mrow></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msub><mi>b</mi><mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>j</mi></mrow></msub></mrow></mfrac></mrow></mrow><mo>.</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(8)</span></td> </tr> </table> <p class="ltx_p">By hypothesis, the LHS of (8) is obviously an integer. However, we have proved already that the last inequality of theorem 1 requires that <math id="S0.SS2.p4.m8" class="ltx_Math" alttext="b_{n+i}<a_{n+i}-1" display="inline"><mrow><msub><mi>b</mi><mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub><mo><</mo><mrow><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub><mo>-</mo><mn>1</mn></mrow></mrow></math> for infinitely many <math id="S0.SS2.p4.m9" class="ltx_Math" alttext="i" display="inline"><mi>i</mi></math>, so that, from the RHS of (8) and the lemma we see that</p> <table id="S0.Ex22" 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.Ex22.m1" class="ltx_Math" alttext="\sum_{i=1}^{\infty}\frac{b_{n+i}}{\prod_{j=1}^{i}a_{n+j}}<\sum_{i=1}^{\infty}% \frac{a_{n+i}-1}{\prod_{j=1}^{i}a_{n+j}}=1," display="block"><mrow><mrow><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><msub><mi>b</mi><mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>j</mi></mrow></msub></mrow></mfrac></mrow><mo><</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mrow><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub><mo>-</mo><mn>1</mn></mrow><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mi>j</mi></mrow></msub></mrow></mfrac></mrow><mo>=</mo><mn>1</mn></mrow><mo>,</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </table> <p class="ltx_p">a clear contradiction, proving the theorem. ∎</p> </div> </div> </section> <section id="S0.SS3" class="ltx_subsection"> <h2 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">0.3 </span>Example</h2> <div id="S0.SS3.p1" class="ltx_para"> <p class="ltx_p"><math id="S0.SS3.p1.m1" class="ltx_Math" alttext="e" display="inline"><mi>e</mi></math> is <em class="ltx_emph ltx_font_italic">irrational</em>. <br class="ltx_break">All we know there are different ways to prove the irrationality of <math id="S0.SS3.p1.m2" class="ltx_Math" alttext="e" display="inline"><mi>e</mi></math>. In particular, it results illustrative if we adapt our theorems and its hypotheses (all of which are true in this case) to this problem. From Taylor’s expansion</p> <table id="S0.Ex23" 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.Ex23.m1" class="ltx_Math" alttext="e=\sum_{i=0}^{\infty}\frac{1}{i!}." display="block"><mrow><mrow><mi>e</mi><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mn>1</mn><mrow><mi>i</mi><mo lspace="0pt" rspace="3.5pt">!</mo></mrow></mfrac></mrow></mrow><mo>.</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </table> <p class="ltx_p">Let us use (1), by setting <math id="S0.SS3.p1.m3" class="ltx_Math" alttext="\rho=e" display="inline"><mrow><mi>ρ</mi><mo>=</mo><mi>e</mi></mrow></math>, <math id="S0.SS3.p1.m4" class="ltx_Math" alttext="b_{0}=2" display="inline"><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>=</mo><mn>2</mn></mrow></math>, <math id="S0.SS3.p1.m5" class="ltx_Math" alttext="b_{i-1}=1" display="inline"><mrow><msub><mi>b</mi><mrow><mi>i</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn></mrow></math> for all <math id="S0.SS3.p1.m6" class="ltx_Math" alttext="i\geq 2" display="inline"><mrow><mi>i</mi><mo>≥</mo><mn>2</mn></mrow></math>, and <math id="S0.SS3.p1.m7" class="ltx_Math" alttext="a_{j}=j+1" display="inline"><mrow><msub><mi>a</mi><mi>j</mi></msub><mo>=</mo><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></mrow></math>, <math id="S0.SS3.p1.m8" class="ltx_Math" alttext="0\leq j\leq i-1" display="inline"><mrow><mn>0</mn><mo>≤</mo><mi>j</mi><mo>≤</mo><mrow><mi>i</mi><mo>-</mo><mn>1</mn></mrow></mrow></math>. Thus,</p> <table id="S0.Ex24" 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.Ex24.m1" class="ltx_Math" alttext="e=2+\sum_{i=1}^{\infty}\frac{1}{\prod_{j=1}^{i}j}." display="block"><mrow><mrow><mi>e</mi><mo>=</mo><mrow><mn>2</mn><mo>+</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mn>1</mn><mrow><msubsup><mo largeop="true" symmetric="true">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><mi>j</mi></mrow></mfrac></mrow></mrow></mrow><mo>.</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </table> <p class="ltx_p">So far our discussion on real numbers. An interesting approach on <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">transcendental numbers</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/TranscendentalNumber.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/transcendentalnumber"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> as well as and extensive bibliography on real numbers are given, for <a class="nnexus_concept" href="http://planetmath.org/substitutionsinpropositionallogic">instance</a>, in <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib1" title="" class="ltx_ref">1</a>]</cite>. <br class="ltx_break"></p> </div> </section> <section id="bib" class="ltx_bibliography"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li id="bib.bib1" class="ltx_bibitem"> <span class="ltx_bibtag ltx_role_refnum">1</span> <span class="ltx_bibblock"> I. Niven, <em class="ltx_emph ltx_font_italic">IRRATIONAL NUMBERS</em>, Ch. VII, pp. 83-88; also p. 157, The <a class="nnexus_concept" href="http://planetmath.org/mathematicalassociationofamerica">Mathematical Association of America</a>, 2005. </span> </li> </ul> </section> <div id="p2" class="ltx_para ltx_align_right"> <table class="ltx_tabular 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"><a class="nnexus_concept" href="http://planetmath.org/representationofrealnumbers">representation of real numbers</a></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">RepresentationOfRealNumbers</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 19:10:36</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 19:10:36</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">perucho (2192)</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">perucho (2192)</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">10</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">perucho (2192)</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">Topic</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 11A63</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l">Related topic</th> <td class="ltx_td ltx_align_left ltx_border_b ltx_border_r">UniquenessOfDigitalRepresentation</td> </tr> </tbody> </table> </div> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Fri Feb 9 15:23:43 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>