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

<b>let </b><font color="Maroon">X</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><a NAME="E1:32"/>
<font color="Maroon">s</font> is <a href="lopban_3.html#V1" target="_self">summable</a>
 ;<br/>

<a NAME="E2:32"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="lopban_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font> is <a href="normsp_1.html#V4">convergent</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:32"/><b>then </b><i><font color="Green">E33</font></i>: 
( <span class="p1">(<span class="default"><a href="lopban_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="lopban_3.html#K3" target="_self">^\</a> 1 is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="lopban_3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><a href="lopban_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>then </b><i><font color="Green">E34</font></i>:  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="lopban_3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><a href="lopban_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="normsp_1.html#K8">lim</a> <span class="p2">(<span class="default"><a href="lopban_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K8">lim</a> <span class="p2">(<span class="default"><a href="lopban_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D1" target="_self">Def1</a>, <a class="ref" href="normsp_1.html#T43">NORMSP_1:43</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
;<br/>
<a NAME="E5:32"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">s</font> <a href="lopban_3.html#K3" target="_self">^\</a> 1 <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="lopban_3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p1">(<span class="default"><a href="lopban_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1"><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/><a NAME="E1:32_2_1"/>
<span class="p1">(<span class="default"><a href="lopban_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E2:32_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="lopban_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</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">s</font> <a href="lopban_3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:32_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="lopban_3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="lopban_3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:32_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="lopban_3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="lopban_3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</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="lopban_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/><a NAME="E5:32_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="lopban_3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="lopban_3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>;<br/><b>hence </b><a NAME="E6:32_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="lopban_3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="lopban_3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</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="E2:32_2"/>
<font color="Maroon">s</font> <a href="lopban_3.html#K3" target="_self">^\</a> 1 <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> <a href="lopban_3.html#K3" target="_self">^\</a> 1</span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p1">(<span class="default"><a href="lopban_3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="normsp_1.html#D6">NORMSP_1:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:32"/><b>then </b>
<font color="Maroon">s</font> <a href="lopban_3.html#K3" target="_self">^\</a> 1 is <a href="normsp_1.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="lopban_3.html#D1" target="_self">Def1</a>, , <a class="ref" href="normsp_1.html#T35">NORMSP_1:35</a></i>;<br/>
<b>hence </b><a NAME="E7:32"/>
( <font color="Maroon">s</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> )
 <b>by </b><i>, , , </i>;<br/>


</div>
