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

<b>let </b><font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>

<b>assume </b><a NAME="E1:93"/><i><font color="Green">E35</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> 1
 ;<br/>

<a NAME="E2:93"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="complex1.html#K6">1r</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="complex1.html#T134">COMPLEX1:134</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  = <font color="Maroon">z</font> <a href="comseq_3.html#K2" target="_self">#N</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<b>consider </b><font color="Maroon">seq1</font> being   <a href="comseq_1.html#NM1">Complex_Sequence</a><b> such that </b><br/><a NAME="E4:93"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq1</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="comseq_1.html#S1">COMSEQ_1:sch 1</a>();<br/></i>
<a NAME="E5:93"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">seq1</font> is <a href="comseq_2.html#V2">convergent</a> &amp;  <a href="comseq_2.html#K2">lim</a> <font color="Maroon">seq1</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a>  )
 
<b>by </b><i>, , <a class="txt" href="comseq_3.html#E2:93"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">seq2</font> = <a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <a href="complex1.html#K6">1r</a>  as   <a href="comseq_1.html#NM1">Complex_Sequence</a> <b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E7:93"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq2</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K6">1r</a> 
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E8:93"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">seq2</font> is <a href="comseq_2.html#V2">convergent</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#D3" target="_self">Def3</a>, <a class="ref" href="comseq_2.html#T9">COMSEQ_2:9</a></i>;<br/>
<a NAME="E9:93"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">seq2</font> <a href="comseq_1.html#K6">-</a> <font color="Maroon">seq1</font> is <a href="comseq_2.html#V2">convergent</a>
 
<b>by </b><i>, <a class="ref" href="comseq_2.html#T25">COMSEQ_2:25</a></i>;<br/>
<i><font color="Green">E47</font></i>:  <a href="comseq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_1.html#K6">-</a> <font color="Maroon">seq1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">seq2</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">seq1</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="comseq_2.html#T26">COMSEQ_2:26</a></i>
<br/>.= 
 <a href="complex1.html#K6">1r</a> 
<b>by </b><i>, <a class="ref" href="comseq_3.html#D3" target="_self">Def3</a>, <a class="ref" href="comseq_2.html#T10">COMSEQ_2:10</a></i>
;<br/>
<a NAME="E11:93"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="comseq_1.html#K4">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_1.html#K6">-</a> <font color="Maroon">seq1</font></span>)</span> is <a href="comseq_2.html#V2">convergent</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#D4" target="_self">Def4</a>, <a class="ref" href="comseq_2.html#T17">COMSEQ_2:17</a></i>;<br/>
<i><font color="Green">E49</font></i>:  <a href="comseq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p3">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="comseq_1.html#K4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_1.html#K6">-</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <a href="complex1.html#K6">1r</a> 
<b>by </b><i><a class="ref" href="comseq_3.html#D4" target="_self">Def4</a>, <a class="ref" href="comseq_3.html#T12" target="_self">Th12</a>, <a class="ref" href="comseq_2.html#T18">COMSEQ_2:18</a></i>
<br/>.= 
<a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a></i>
;<br/>
<div><i><font color="Green">E50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="93_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="comseq_3.html#K2" target="_self">#N</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_1.html#K5">-</a> <font color="Maroon">seq1</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D9">COMSEQ_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_1.html#K2">+</a> <span class="p3">(<span class="default"><a href="comseq_1.html#K5">-</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_1.html#K6">-</a> <font color="Maroon">seq1</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D10">COMSEQ_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p3">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="comseq_1.html#K4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_1.html#K6">-</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="comseq_1.html#D7">COMSEQ_1:def 7</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:93"/>
 <a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span> is <a href="comseq_2.html#V2">convergent</a>
 <b>by </b><i><a class="ref" href="comseq_3.html#T13" target="_self">Th13</a>, <a class="ref" href="comseq_1.html#T6">COMSEQ_1:6</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="comseq_3.html#D10" target="_self">COMSEQ_3:def 10</a><br/>

<a NAME="E15:93"/>
 <a href="comseq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="comseq_3.html#T14" target="_self">Th14</a>, <a class="ref" href="comseq_3.html#T15" target="_self">Th15</a>, <a class="ref" href="comseq_1.html#T6">COMSEQ_1:6</a></i>;<br/>
<b>hence </b><a NAME="E16:93"/>
 <a href="comseq_3.html#K8" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span> <a href="hidden.html#R1">=</a> <a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">z</font></span>)</span>
 ;<br/>


</div>
