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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">S1</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K2" title="NORMSP_2:func.2">MetricSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>;<br/><b>assume </b><a NAME="E1:3_1_1_1"/><i><font color="Green" title="E2">A1</font></i>: 
<font color="Maroon" title="c2">S1</font> is <a href="tbsp_1.html#V3" title="TBSP_1:attr.3">Cauchy</a>
 ;<br/><b>reconsider </b><font color="Maroon" title="c3">S2</font> = <font color="Maroon" title="c2">S1</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E3:3_1_1_1"/>
<font color="Maroon" title="c3">S2</font> is <a href="rsspace3.html#V1" title="RSSPACE3:attr.1">CCauchy</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="rsspace3.html#D5" title="RSSPACE3:def.5">RSSPACE3:def 5</a><br/>

<b>assume </b><a NAME="E1:3_1_1_1_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> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">p</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="E3:3_1_1_1_1"/><i><font color="Green" title="E3">A2</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="Maroon" title="c5">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c5">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">m</font></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="lopban_5.html#E1:3_1_1_1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="tbsp_1.html#D5" title="TBSP_1:def.5">TBSP_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">n</font>, <font color="Maroon" title="c7">m</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/><b>assume </b><a NAME="E1:3_1_1_1_1_1"/>
( <font color="Maroon" title="c5">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c5">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font> )
 ;<br/><a NAME="E2:3_1_1_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">m</font></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="lopban_5.html#E3:3_1_1_1_1"><i><font color="Green" title="E3">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E3:3_1_1_1_1_1"/>
 <a href="rsspace3.html#K4" title="RSSPACE3:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">m</font></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="ref" href="normsp_2.html#D1" title="NORMSP_2:def.1">NORMSP_2:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:3_1_1_1_1"/>
 ex <font color="Olive">b<sub>1</sub></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">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></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>   holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive">b<sub>1</sub></font> or  <a href="rsspace3.html#K4" title="RSSPACE3:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">S2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font> )
 ;<br/>


</div><b>end;</b></div><a NAME="E4:3_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">S2</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a>
 <b>by </b><i><a class="ref" href="lopban_1.html#D16" title="LOPBAN_1:def.16">LOPBAN_1:def 16</a></i>;<br/><b>hence </b><a NAME="E5:3_1_1_1"/>
<font color="Maroon" title="c2">S1</font> is <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a>
 <b>by </b><i><a class="ref" href="normsp_2.html#T5" title="NORMSP_2:th.5">NORMSP_2:5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:3_1_1"/>
 <a href="normsp_2.html#K2" title="NORMSP_2:func.2">MetricSpaceNorm</a> <font color="Maroon" title="c1">X</font> is <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a>
 <b>by </b><i><a class="ref" href="tbsp_1.html#D6" title="TBSP_1:def.6">TBSP_1:def 6</a></i>;<br/>


</div>
