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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="csspace.html#NM1" title="CSSPACE:NM.1">ComplexUnitarySpace</a>;<br/><b>let </b><font color="Maroon" title="c2">seq</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">Cseq</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:59"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="clvect_2.html#V5" title="CLVECT_2:attr.5">Hilbert</a>
 <b>and </b><br/><a NAME="E2:59"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c2">seq</font> is <a href="clvect_2.html#V2" title="CLVECT_2:attr.2">Cauchy</a>
 <b>and </b><br/><a NAME="E3:59"/><i><font color="Green" title="E27">A3</font></i>: 
<font color="Maroon" title="c3">Cseq</font> is <a href="clvect_3.html#V3" title="CLVECT_3:attr.3">Cauchy</a>
 ;<br/>

<a NAME="E4:59"/>
<font color="Maroon" title="c1">X</font> is <a href="clvect_2.html#V4" title="CLVECT_2:attr.4">complete</a>
 
<b>by </b><i><a class="txt" href="clvect_3.html#E1:59"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="clvect_2.html#D13" title="CLVECT_2:def.13">CLVECT_2:def 13</a></i>;<br/>
<a NAME="E5:59"/><b>then </b><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon" title="c2">seq</font> is <a href="clvect_2.html#V1" title="CLVECT_2:attr.1">convergent</a>
 
<b>by </b><i><a class="txt" href="clvect_3.html#E2:59"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="clvect_2.html#D12" title="CLVECT_2:def.12">CLVECT_2:def 12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:59_1"/>
<font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>then consider </b><font color="Maroon" title="c5">k</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> <b> such that </b><br/><a NAME="E3:59_1"/><i><font color="Green" title="E29">A5</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">k</font> holds <br/><span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="txt" href="clvect_3.html#E3:59"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="clvect_3.html#D10" target="_self" title="CLVECT_3:def.10">Def10</a></i>;<br/><b>take </b><font color="Maroon" title="c6">k</font> = <font color="Maroon" title="c5">k</font>;<br/><b>thus </b><a NAME="E4:59_1"/>
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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">k</font> holds <br/><span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="txt" href="clvect_3.html#E3:59_1"><i><font color="Green" title="E29">A5</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:59"/><b>then </b>
<font color="Maroon" title="c3">Cseq</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#T46" title="COMSEQ_3:th.46">COMSEQ_3:46</a></i>;<br/>
<b>hence </b><a NAME="E8:59"/>
<font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <font color="Maroon" title="c2">seq</font> is <a href="clvect_2.html#V2" title="CLVECT_2:attr.2">Cauchy</a>
 <b>by </b><i><a class="txt" href="clvect_3.html#E5:59"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="clvect_3.html#T48" target="_self" title="CLVECT_3:th.48">Th48</a>, <a class="ref" href="clvect_2.html#T65" title="CLVECT_2:th.65">CLVECT_2:65</a></i>;<br/>


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