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

<b>let </b><font color="Maroon" title="c1">seq</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:7"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>
 ;<br/>

<a NAME="E2:7"/><b>then </b><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#T64" title="COMSEQ_3:th.64">COMSEQ_3:64</a></i>;<br/>
<a NAME="E3:7"/><b>then </b><i><font color="Green" title="E10">A3</font></i>: 
 <a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <font color="Maroon" title="c1">seq</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#D10" title="COMSEQ_3:def.10">COMSEQ_3:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c2">Pseq</font> =  <a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <font color="Maroon" title="c1">seq</font> as  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> <b>by </b><i><a class="txt" href="csspace2.html#E2:7"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="comseq_3.html#D10" title="COMSEQ_3:def.10">COMSEQ_3:def 10</a></i>;<br/>
<a NAME="E5:7"/><i><font color="Green" title="E11">A4</font></i>: 
( <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">Pseq</font></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">Pseq</font></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <font color="Maroon" title="c1">seq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> )
 
;<br/>
<i><font color="Green" title="E12">A5</font></i>: <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <font color="Maroon" title="c1">seq</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> = 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <font color="Maroon" title="c1">seq</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_3.html#D8" title="COMSEQ_3:def.8">COMSEQ_3:def 8</a></i>
<br/>.= 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <font color="Maroon" title="c1">seq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span>
<b>by </b><i><a class="txt" href="csspace2.html#E3:7"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="comseq_2.html#T11" title="COMSEQ_2:th.11">COMSEQ_2:11</a></i>
;<br/>
<a NAME="E7:7"/>
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><font color="Maroon" title="c1">seq</font></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<b>by </b><i><a class="txt" href="csspace2.html#E1:7"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="comseq_3.html#D11" title="COMSEQ_3:def.11">COMSEQ_3:def 11</a></i>;<br/>
<a NAME="E8:7"/><b>then </b><i><font color="Green" title="E13">A6</font></i>: 
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><font color="Maroon" title="c1">seq</font></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="series_1.html#D2" title="SERIES_1:def.2">SERIES_1:def 2</a></i>;<br/>
<a NAME="E9:7"/>
for <font color="Olive" title="b1">k</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  <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <font color="Maroon" title="c1">seq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><font color="Maroon" title="c1">seq</font></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">k</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/>

<a NAME="E1:7_2"/>
<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"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><font color="Maroon" title="c1">seq</font></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c3">k</font>
 
<b>by </b><i><a class="ref" href="comseq_3.html#T30" title="COMSEQ_3:th.30">COMSEQ_3:30</a></i>;<br/>
<b>hence </b><a NAME="E2:7_2"/>
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <font color="Maroon" title="c1">seq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><font color="Maroon" title="c1">seq</font></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="ref" href="comseq_1.html#D14" title="COMSEQ_1:def.14">COMSEQ_1:def 14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:7"/><b>then </b>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <font color="Maroon" title="c1">seq</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><font color="Maroon" title="c1">seq</font></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="csspace2.html#E5:7"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="csspace2.html#E8:7"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="seq_2.html#T32" title="SEQ_2:th.32">SEQ_2:32</a></i>;<br/>
<b>hence </b><a NAME="E11:7"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <font color="Maroon" title="c1">seq</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><font color="Maroon" title="c1">seq</font></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span>
 <b>by </b><i><a class="txt" href="csspace2.html#E6:7"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="series_1.html#D3" title="SERIES_1:def.3">SERIES_1:def 3</a></i>;<br/>


</div>
