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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="csspace.html#NM1">ComplexUnitarySpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>assume </b><a NAME="E1:12"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="clvect_3.html#V1" target="_self">summable</a>
 ;<br/>

<a NAME="E2:12"/><b>then </b><i><font color="Green">E7</font></i>: 
 <a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font> is <a href="clvect_2.html#V1">convergent</a>
 
<b>by </b><i><a class="ref" href="clvect_3.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E3:12"/><b>then </b><i><font color="Green">E8</font></i>: 
( <span class="p1">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="clvect_2.html#K8">^\</a> 1 is <a href="clvect_2.html#V1">convergent</a> &amp;  <a href="clvect_2.html#K1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="clvect_2.html#K8">^\</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="clvect_2.html#K1">lim</a> <span class="p1">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="clvect_2.html#T100">CLVECT_2:100</a></i>;<br/>
<b>then </b><i><font color="Green">E9</font></i>:  <a href="clvect_2.html#K1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="clvect_2.html#K8">^\</a> 1</span>)</span> <a href="clvect_1.html#K7">-</a> <span class="p2">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="clvect_2.html#K1">lim</a> <span class="p2">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="clvect_2.html#K1">lim</a> <span class="p2">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="clvect_3.html#E2:12"><i><font color="Green">E7</font></i></a>, <a class="ref" href="clvect_2.html#T14">CLVECT_2:14</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>)
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
;<br/>
<a NAME="E5:12"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="clvect_2.html#K8">^\</a> 1 <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="clvect_2.html#K8">^\</a> 1</span>)</span> <a href="clvect_1.html#K7">-</a> <span class="p1">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><span class="p1">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clvect_3.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clvect_2.html#K8">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_2.html#D11">CLVECT_2:def 11</a></i>
;<br/><b>hence </b><a NAME="E2:12_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="clvect_2.html#K8">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clvect_2.html#K8">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="clvect_2.html#D11">CLVECT_2:def 11</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:12_2"/><b>then </b><i><font color="Green">E11</font></i>: 
<span class="p1">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="clvect_2.html#K8">^\</a> 1 <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="csspace.html#K18">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clvect_2.html#K8">^\</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="clvect_1.html#D11">CLVECT_1:def 11</a></i>;<br/>
<a NAME="E3:12_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clvect_2.html#K8">^\</a> 1</span>)</span> <a href="csspace.html#K18">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="clvect_1.html#K7">-</a> <span class="p2">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="clvect_2.html#K8">^\</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clvect_2.html#K8">^\</a> 1</span>)</span> <a href="csspace.html#K18">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="clvect_1.html#K7">-</a> <span class="p3">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clvect_2.html#K8">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="clvect_1.html#K7">-</a> <span class="p3">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D11">CLVECT_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clvect_2.html#K8">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D12">CLVECT_1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clvect_2.html#K8">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>)
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clvect_2.html#K8">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:12_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clvect_2.html#K8">^\</a> 1</span>)</span> <a href="csspace.html#K18">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="clvect_1.html#K7">-</a> <span class="p2">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="clvect_2.html#K8">^\</a> 1
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:12_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="clvect_2.html#K8">^\</a> 1 <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="clvect_2.html#K8">^\</a> 1</span>)</span> <a href="clvect_1.html#K7">-</a> <span class="p1">(<span class="default"><a href="clvect_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="clvect_3.html#E2:12_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="csspace.html#T83">CSSPACE:83</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:12"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="clvect_2.html#K8">^\</a> 1 is <a href="clvect_2.html#V1">convergent</a>
 
<b>by </b><i><a class="txt" href="clvect_3.html#E2:12"><i><font color="Green">E7</font></i></a>, <a class="txt" href="clvect_3.html#E3:12"><i><font color="Green">E8</font></i></a>, <a class="ref" href="clvect_2.html#T4">CLVECT_2:4</a></i>;<br/>
<b>hence </b><a NAME="E7:12"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="clvect_2.html#V1">convergent</a>
 <b>by </b><i><a class="ref" href="clvect_2.html#T101">CLVECT_2:101</a></i>;<br/>

<a NAME="E8:12"/>
<font color="Maroon">c<sub>2</sub></font> <a href="clvect_2.html#K8">^\</a> 1 is    <a href="bhsp_3.html#M1">subsequence</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="clvect_2.html#T99">CLVECT_2:99</a></i>;<br/>
<b>hence </b><a NAME="E9:12"/>
 <a href="clvect_2.html#K1">lim</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="clvect_3.html#E4:12"><i><font color="Green">E9</font></i></a>, <a class="txt" href="clvect_3.html#E5:12"><i><font color="Green">E10</font></i></a>, <a class="txt" href="clvect_3.html#E7:12"><i><font color="Green">E11</font></i></a>, <a class="ref" href="clvect_2.html#T89">CLVECT_2:89</a></i>;<br/>


</div>
