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<span class="kw">let </span><font color="Maroon" title="c1">Cseq</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font> being   <a href="csspace2.html#NM1" title="CSSPACE2:NM.1">ComplexHilbertSpace</a><br/>  for <font color="Olive" title="b2">seq</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Olive" title="b1">X</font>  st <font color="Olive" title="b2">seq</font> is  <a href="clvect_2.html#V2" title="CLVECT_2:attr.2">Cauchy</a>  &amp; <font color="Maroon" title="c1">Cseq</font> is  <a href="clvect_3.html#V3" title="CLVECT_3:attr.3">Cauchy</a>  holds <br/><font color="Maroon" title="c1">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <font color="Olive" title="b2">seq</font> is  <a href="clvect_2.html#V2" title="CLVECT_2:attr.2">Cauchy</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font> be   <a href="csspace2.html#NM1" title="CSSPACE2:NM.1">ComplexHilbertSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">seq</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b1">seq</font> is  <a href="clvect_2.html#V2" title="CLVECT_2:attr.2">Cauchy</a>  &amp; <font color="Maroon" title="c1">Cseq</font> is  <a href="clvect_3.html#V3" title="CLVECT_3:attr.3">Cauchy</a>  holds <br/><font color="Maroon" title="c1">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <font color="Olive" title="b1">seq</font> is  <a href="clvect_2.html#V2" title="CLVECT_2:attr.2">Cauchy</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">seq</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">seq</font> is  <a href="clvect_2.html#V2" title="CLVECT_2:attr.2">Cauchy</a>  &amp; <font color="Maroon" title="c1">Cseq</font> is  <a href="clvect_3.html#V3" title="CLVECT_3:attr.3">Cauchy</a>  implies <font color="Maroon" title="c1">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <font color="Maroon" title="c3">seq</font> is  <a href="clvect_2.html#V2" title="CLVECT_2:attr.2">Cauchy</a>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:57"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c3">seq</font> is  <a href="clvect_2.html#V2" title="CLVECT_2:attr.2">Cauchy</a> 
 <span class="kw">and </span><br/><a NAME="E2:57"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
<font color="Maroon" title="c1">Cseq</font> is  <a href="clvect_3.html#V3" title="CLVECT_3:attr.3">Cauchy</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <font color="Maroon" title="c3">seq</font> is  <a href="clvect_2.html#V2" title="CLVECT_2:attr.2">Cauchy</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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>  implies  ex <font color="Olive" title="b1">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>  st <br/> for <font color="Olive" title="b2">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="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">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="c1">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> )</span><br/><span class="kw">assume </span><a NAME="E1:57_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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>  st <br/> for <font color="Olive" title="b2">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="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">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="c1">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font></span><br/><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:57_1"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
 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="c1">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c1">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="clvect_3.html#D9" target="_self" title="CLVECT_3:def.9">Def10</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c6">k</font> = <font color="Maroon" title="c5">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="c1">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c1">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font></span><br/><span class="kw">thus </span><a NAME="E4:57_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="c1">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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="c1">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_1"><span class="lab"><font color="Green" title="E27">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:57"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A4</font></span>: 
<font color="Maroon" title="c1">Cseq</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#T46" title="COMSEQ_3:th.46">COMSEQ_3:46</a></span>;<br/>
<a NAME="E5:57"/>
<font color="Maroon" title="c3">seq</font> is  <a href="clvect_2.html#V1" title="CLVECT_2:attr.1">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="clvect_2.html#D11" title="CLVECT_2:def.11">CLVECT_2:def 11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:57"/>
<font color="Maroon" title="c1">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <font color="Maroon" title="c3">seq</font> is  <a href="clvect_2.html#V2" title="CLVECT_2:attr.2">Cauchy</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E27">A4</font></span></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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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