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<b>let </b><font color="Maroon">X</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</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">Rseq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:54"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">X</font> <a href="bhsp_3.html#NR5" target="_self">is_Hilbert_space</a> 
 <b>and </b><br/><a NAME="E2:54"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">seq</font> <a href="bhsp_3.html#NR1" target="_self">is_Cauchy_sequence</a> 
 <b>and </b><br/><a NAME="E3:54"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">Rseq</font> <a href="bhsp_4.html#NR1" target="_self">is_Cauchy_sequence</a> 
 ;<br/>

<a NAME="E4:54"/>
<font color="Maroon">X</font> <a href="bhsp_3.html#NR4" target="_self">is_complete_space</a> 
 
<b>by </b><i>, <a class="ref" href="bhsp_3.html#D7">BHSP_3:def 7</a></i>;<br/>
<a NAME="E5:54"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">seq</font> is <a href="bhsp_2.html#V1">convergent</a>
 
<b>by </b><i>, <a class="ref" href="bhsp_3.html#D6">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">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><a NAME="E1:54_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">r</font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>assume </b><a NAME="E2:54_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="E4:54_1"/><i><font color="Green">E59</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/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&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="E5:54_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/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E7:54"/><b>then </b>
<font color="Maroon">Rseq</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T58">SEQ_4:58</a></i>;<br/>
<a NAME="E8:54"/><b>then </b>
<font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">seq</font> is <a href="bhsp_2.html#V1">convergent</a>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:54"/>
<font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">seq</font> <a href="bhsp_3.html#NR1" target="_self">is_Cauchy_sequence</a> 
 <b>by </b><i><a class="ref" href="bhsp_3.html#T9">BHSP_3:9</a></i>;<br/>


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