<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:35"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="normsp_1.html#NM2">sequence</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="bhsp_3.html#NR1" target="_self">is_Cauchy_sequence</a>  holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="bhsp_2.html#V1">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">c<sub>2</sub></font> be   <a href="normsp_1.html#NM2">sequence</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:35_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="bhsp_3.html#NR1" target="_self">is_Cauchy_sequence</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="normsp_1.html#NM2">sequence</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="real_ns1.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E3:35_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="real_ns1.html#D6" target="_self">Def6</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E5:35_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="real_ns1.html#D4" target="_self">Def4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E7:35_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="rsspace3.html#V1">CCauchy</a>
 
<b>by </b><i><a class="txt" href="real_ns1.html#E1:35_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="real_ns1.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E8:35_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="normsp_1.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="real_ns1.html#T12" target="_self">Th12</a></i>;<br/>
<b>hence </b><a NAME="E9:35_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="bhsp_2.html#V1">convergent</a>
 <b>by </b><i><a class="ref" href="real_ns1.html#T16" target="_self">Th16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:35"/><b>then </b>
 <a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">c<sub>1</sub></font> <a href="bhsp_3.html#NR4" target="_self">is_complete_space</a> 
 
<b>by </b><i><a class="ref" href="bhsp_3.html#D6">BHSP_3:def 6</a></i>;<br/>
<b>hence </b><a NAME="E3:35"/>
 <a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">c<sub>1</sub></font> is <a href="bhsp_3.html#V4">Hilbert</a>
 <b>by </b><i><a class="ref" href="bhsp_3.html#D7">BHSP_3:def 7</a></i>;<br/>


</div>
