<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>let </b><font color="Maroon" title="c2">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:40"/><i><font color="Green" title="E30">A1</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">r</font>
 <b>and </b><br/><a NAME="E2:40"/><i><font color="Green" title="E31">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">r</font></span>)</span>
 ;<br/>

<a NAME="E3:40"/><i><font color="Green" title="E32">A3</font></i>: 
<font color="Maroon" title="c2">seq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="seq_4.html#E1:40"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="seq_4.html#E2:40"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="seq_4.html#T47" target="_self" title="SEQ_4:th.47">Th47</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">p</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:40_1"/><i><font color="Green" title="E33">A4</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font>
 ;<br/><a NAME="E2:40_1"/><b>then </b><i><font color="Green" title="E34">A5</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T124" title="XREAL_1:th.124">XREAL_1:124</a></i>;<br/><b>consider </b><font color="Maroon" title="c4">k1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:40_1"/><i><font color="Green" title="E35">A6</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k1</font>
 <b>by </b><i><a class="ref" href="seq_4.html#T10" target="_self" title="SEQ_4:th.10">Th10</a></i>;<br/><b>take </b><font color="Maroon" title="c5">n</font> = <font color="Maroon" title="c4">k1</font>;<br/><b>let </b><font color="Maroon" title="c6">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E5:40_1"/><i><font color="Green" title="E36">A7</font></i>: 
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font>
 ;<br/><a NAME="E6:40_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E7:40_1"/><b>then </b><i><font color="Green" title="E37">A8</font></i>: 
<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E5:40_1"><i><font color="Green" title="E36">A7</font></i></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></i>;<br/><a NAME="E8:40_1"/><i><font color="Green" title="E38">A9</font></i>: 
<font color="Maroon" title="c4">k1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="seq_4.html#E1:40_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="seq_4.html#E4:40_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="xreal_1.html#T124" title="XREAL_1:th.124">XREAL_1:124</a></i>;<br/><a NAME="E9:40_1"/><b>then </b>
<font color="Maroon" title="c4">k1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E10:40_1"/><b>then </b>
<font color="Maroon" title="c4">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">k1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T153" title="XREAL_1:th.153">XREAL_1:153</a></i>;<br/><a NAME="E11:40_1"/><b>then </b><i><font color="Green" title="E39">A10</font></i>: 
<font color="Maroon" title="c4">k1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">k1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E12:40_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E1:40"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="seq_4.html#E4:40_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E13:40_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">k1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E11:40_1"><i><font color="Green" title="E39">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E14:40_1"/><b>then </b>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">k1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="seq_4.html#E2:40_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="xreal_1.html#T78" title="XREAL_1:th.78">XREAL_1:78</a></i>;<br/><a NAME="E15:40_1"/><b>then </b><i><font color="Green" title="E40">A11</font></i>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">k1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>;<br/><a NAME="E16:40_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 <b>by </b><i><a class="txt" href="seq_4.html#E8:40_1"><i><font color="Green" title="E38">A9</font></i></a>, <a class="ref" href="square_1.html#T74" title="SQUARE_1:th.74">SQUARE_1:74</a></i>;<br/><a NAME="E17:40_1"/><b>then </b><i><font color="Green" title="E41">A12</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E1:40"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E18:40_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E7:40_1"><i><font color="Green" title="E37">A8</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E19:40_1"/><b>then </b>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="seq_4.html#E17:40_1"><i><font color="Green" title="E41">A12</font></i></a>, <a class="ref" href="xreal_1.html#T120" title="XREAL_1:th.120">XREAL_1:120</a></i>;<br/><a NAME="E20:40_1"/><b>then </b><i><font color="Green" title="E42">A13</font></i>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E15:40_1"><i><font color="Green" title="E40">A11</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E21:40_1"/><i><font color="Green" title="E43">A14</font></i>: 
<font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="seq_4.html#E2:40"><i><font color="Green" title="E31">A2</font></i></a></i>;<br/><a NAME="E22:40_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E23:40_1"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E1:40"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E24:40_1"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></i>;<br/><b>hence </b><a NAME="E25:40_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E20:40_1"><i><font color="Green" title="E42">A13</font></i></a>, <a class="txt" href="seq_4.html#E21:40_1"><i><font color="Green" title="E43">A14</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:40"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="seq_4.html#E3:40"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></i>;<br/>


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