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<a NAME="E1:20_1_1"/>
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c3">s</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c4">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <b> such that </b><br/><a NAME="E2:20_1_1_1"/><i><font color="Green" title="E12">A1</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <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="Olive" title="b1">p</font> holds <br/> ex <font color="Olive" title="b2">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>  st <br/>for <font color="Olive" title="b3">m</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>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/><span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font>
 <b>by </b><i><a class="ref" href="comseq_2.html#D4" target="_self" title="COMSEQ_2:def.4">Def4</a></i>;<br/>
<b>take </b><font color="Maroon" title="c5">R</font> = <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">g</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_2.html#D6" title="SEQ_2:def.6">SEQ_2:def 6</a><br/>

<b>let </b><font color="Maroon" title="c6">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/>

<a NAME="E3:20_1_1_1"/><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c6">p</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>assume </b><a NAME="E4:20_1_1_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="c6">p</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">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> <b> such that </b><br/><a NAME="E6:20_1_1_1"/><i><font color="Green" title="E14">A3</font></i>: 
for <font color="Olive" title="b1">m</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>   st <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="txt" href="comseq_2.html#E2:20_1_1_1"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="comseq_2.html#E3:20_1_1_1"><i><font color="Green" title="E13">A2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">n</font>
;<br/>

<b>let </b><font color="Maroon" title="c8">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="E7:20_1_1_1"/>
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font>
 ;<br/>

<a NAME="E8:20_1_1_1"/><b>then </b>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font>
 
<b>by </b><i><a class="txt" href="comseq_2.html#E6:20_1_1_1"><i><font color="Green" title="E14">A3</font></i></a></i>;<br/>
<a NAME="E9:20_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">g</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 
<b>by </b><i><a class="ref" href="complex1.html#T150" title="COMPLEX1:th.150">COMPLEX1:150</a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E10:20_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">g</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<b>hence </b><a NAME="E11:20_1_1_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c3">s</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="ref" href="valued_1.html#T18" title="VALUED_1:th.18">VALUED_1:18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:20_1_1"/>
<span class="p1"><a href="valued_1.html#K54" title="VALUED_1:func.54">|.</a><span class="default"><font color="Maroon" title="c3">s</font></span><a href="valued_1.html#K54" title="VALUED_1:func.54">.|</a></span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>
 ;<br/>


</div>
