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<b>let </b><font color="Maroon">X</font> be   <a href="clopban2.html#NM3">Complex_Banach_Algebra</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>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/>





<b>assume </b><a NAME="E1:5"/><i><font color="Green">E51</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">s</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/>

<div><i><font color="Green">E52</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">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:5_1"/><i><font color="Green">E53</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/><b>take </b><font color="Maroon">k</font> = 0;<br/><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>assume </b><a NAME="E2:5_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E3:5_1"/>
<font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:5_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">z</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="clvect_1.html#T108">CLVECT_1:108</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:5"/><b>then </b>
<font color="Maroon">s</font> is <a href="clvect_1.html#V6">convergent</a>
 
<b>by </b><i><a class="ref" href="clvect_1.html#D15">CLVECT_1:def 15</a></i>;<br/>
<b>hence </b><a NAME="E4:5"/>
 <a href="clvect_1.html#K11">lim</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="clvect_1.html#D17">CLVECT_1:def 17</a></i>;<br/>


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