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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon" title="c2">seq</font> be   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">Rseq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:54"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="bhsp_3.html#NR5" title="BHSP_3:NR.5">is_Hilbert_space</a> 
 <b>and </b><br/><a NAME="E2:54"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#NR1" title="BHSP_3:NR.1">is_Cauchy_sequence</a> 
 <b>and </b><br/><a NAME="E3:54"/><i><font color="Green" title="E29">A3</font></i>: 
<font color="Maroon" title="c3">Rseq</font> <a href="bhsp_4.html#NR1" title="BHSP_4:NR.1">is_Cauchy_sequence</a> 
 ;<br/>

<a NAME="E4:54"/>
<font color="Maroon" title="c1">X</font> <a href="bhsp_3.html#NR4" title="BHSP_3:NR.4">is_complete_space</a> 
 
<b>by </b><i><a class="txt" href="bhsp_4.html#E1:54"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="bhsp_3.html#D7" title="BHSP_3:def.7">BHSP_3:def 7</a></i>;<br/>
<a NAME="E5:54"/><b>then </b><i><font color="Green" title="E30">A4</font></i>: 
<font color="Maroon" title="c2">seq</font> is <a href="bhsp_2.html#V1" title="BHSP_2:attr.1">convergent</a>
 
<b>by </b><i><a class="txt" href="bhsp_4.html#E2:54"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="bhsp_3.html#D6" title="BHSP_3:def.6">BHSP_3:def 6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><a NAME="E1:54_1"/><i><font color="Green" title="E31">A5</font></i>: 
<font color="Maroon" title="c4">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><b>assume </b><a NAME="E2:54_1"/>
<font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<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="E4:54_1"/><i><font color="Green" title="E32">A6</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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="txt" href="bhsp_4.html#E3:54"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="bhsp_4.html#E1:54_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="bhsp_4.html#D10" target="_self" title="BHSP_4: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="E5:54_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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="txt" href="bhsp_4.html#E4:54_1"><i><font color="Green" title="E32">A6</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:54"/><b>then </b>
<font color="Maroon" title="c3">Rseq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T58" title="SEQ_4:th.58">SEQ_4:58</a></i>;<br/>
<a NAME="E8:54"/><b>then </b>
<font color="Maroon" title="c3">Rseq</font> <a href="bhsp_4.html#K5" title="BHSP_4:func.5">*</a> <font color="Maroon" title="c2">seq</font> is <a href="bhsp_2.html#V1" title="BHSP_2:attr.1">convergent</a>
 
<b>by </b><i><a class="txt" href="bhsp_4.html#E5:54"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="bhsp_4.html#T50" target="_self" title="BHSP_4:th.50">Th50</a></i>;<br/>
<b>hence </b><a NAME="E9:54"/>
<font color="Maroon" title="c3">Rseq</font> <a href="bhsp_4.html#K5" title="BHSP_4:func.5">*</a> <font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#NR1" title="BHSP_3:NR.1">is_Cauchy_sequence</a> 
 <b>by </b><i><a class="ref" href="bhsp_3.html#T9" title="BHSP_3:th.9">BHSP_3:9</a></i>;<br/>


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