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<b>let </b><font color="Maroon">X</font> be   <a href="csspace.html#NM1">ComplexUnitarySpace</a>;<br/><b>let </b><font color="Maroon">seq</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">Cseq</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:60"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">X</font> is <a href="clvect_2.html#V5">Hilbert</a>
 <b>and </b><br/><a NAME="E2:60"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">seq</font> is <a href="clvect_2.html#V2">Cauchy</a>
 <b>and </b><br/><a NAME="E3:60"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">Cseq</font> is <a href="clvect_3.html#V3" target="_self">Cauchy</a>
 ;<br/>

<a NAME="E4:60"/>
<font color="Maroon">X</font> is <a href="clvect_2.html#V4">complete</a>
 
<b>by </b><i>, <a class="ref" href="clvect_2.html#D13">CLVECT_2:def 13</a></i>;<br/>
<a NAME="E5:60"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">seq</font> is <a href="clvect_2.html#V1">convergent</a>
 
<b>by </b><i>, <a class="ref" href="clvect_2.html#D12">CLVECT_2:def 12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:60_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:60_1"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">m</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/><b>take </b><font color="Maroon">k</font> = <font color="Maroon">k</font>;<br/><b>thus </b><a NAME="E4:60_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E7:60"/><b>then </b>
<font color="Maroon">Cseq</font> is <a href="comseq_2.html#V2">convergent</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#T46">COMSEQ_3:46</a></i>;<br/>
<a NAME="E8:60"/><b>then </b>
<font color="Maroon">Cseq</font> <a href="clvect_3.html#K7" target="_self">*</a> <font color="Maroon">seq</font> is <a href="clvect_2.html#V1">convergent</a>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:60"/>
<font color="Maroon">Cseq</font> <a href="clvect_3.html#K7" target="_self">*</a> <font color="Maroon">seq</font> is <a href="clvect_2.html#V2">Cauchy</a>
 <b>by </b><i><a class="ref" href="clvect_2.html#T65">CLVECT_2:65</a></i>;<br/>


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