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<b>let </b><font color="Maroon" title="c1">n</font> be    <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> ;<br/>

<a NAME="E1:35"/>
for <font color="Olive" title="b1">seq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">seq</font> <a href="bhsp_3.html#NR1" title="BHSP_3:NR.1">is_Cauchy_sequence</a>  holds <br/><font color="Olive" title="b1">seq</font> is <a href="bhsp_2.html#V1" title="BHSP_2:attr.1">convergent</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<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 <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:35_1"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#NR1" title="BHSP_3:NR.1">is_Cauchy_sequence</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c3">seq'</font> = <font color="Maroon" title="c2">seq</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="ref" href="real_ns1.html#T15" target="_self" title="REAL_NS1:th.15">Th15</a></i>;<br/>
<a NAME="E3:35_1"/>
<font color="Maroon" title="c3">seq'</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="real_ns1.html#E1:35_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="real_ns1.html#T12" target="_self" title="REAL_NS1:th.12">Th12</a>, <a class="ref" href="real_ns1.html#T18" target="_self" title="REAL_NS1:th.18">Th18</a></i>;<br/>
<b>hence </b><a NAME="E4:35_1"/>
<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="ref" href="real_ns1.html#T16" target="_self" title="REAL_NS1:th.16">Th16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:35"/><b>then </b>
 <a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font> <a href="bhsp_3.html#NR4" title="BHSP_3:NR.4">is_complete_space</a> 
 
<b>by </b><i><a class="ref" href="bhsp_3.html#D6" title="BHSP_3:def.6">BHSP_3:def 6</a></i>;<br/>
<b>hence </b><a NAME="E3:35"/>
 <a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font> is <a href="bhsp_3.html#V4" title="BHSP_3:attr.4">Hilbert</a>
 <b>by </b><i><a class="ref" href="bhsp_3.html#D7" title="BHSP_3:def.7">BHSP_3:def 7</a></i>;<br/>


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