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

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="clopban2.html#NM3" title="CLOPBAN2:NM.3">Complex_Banach_Algebra</a>;<br/><b>let </b><font color="Maroon" title="c2">s</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>





<b>assume </b><a NAME="E1:5"/><i><font color="Green" title="E5">A1</font></i>: 
for <font color="Olive" title="b1">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>  holds  <font color="Maroon" title="c2">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font>
 ;<br/>

<div><i><font color="Green" title="E6">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:5_1"/><i><font color="Green" title="E7">A3</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">p</font>
 ;<br/><b>take </b><font color="Maroon" title="c5">k</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/><b>let </b><font color="Maroon" title="c6">n</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="E2:5_1"/>
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font>
 ;<br/><a NAME="E3:5_1"/>
<font color="Maroon" title="c2">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font>
 <b>by </b><i><a class="txt" href="clopban4.html#E1:5"><i><font color="Green" title="E5">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E4:5_1"/>
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">z</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="txt" href="clopban4.html#E1:5_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="clvect_1.html#T108" title="CLVECT_1:th.108">CLVECT_1:108</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:5"/><b>then </b>
<font color="Maroon" title="c2">s</font> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a>
 
<b>by </b><i><a class="ref" href="clvect_1.html#D16" title="CLVECT_1:def.16">CLVECT_1:def 16</a></i>;<br/>
<b>hence </b><a NAME="E4:5"/>
 <a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font>
 <b>by </b><i><a class="txt" href="clopban4.html#E2:5"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="clvect_1.html#D18" title="CLVECT_1:def.18">CLVECT_1:def 18</a></i>;<br/>


</div>
