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

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#NM1">MetrSpace</a>;<br/><b>let </b><font color="Maroon">S</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><a NAME="E1:35"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">S</font> is <a href="tbsp_1.html#V3">Cauchy</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">r</font> = 1;<br/><b>thus </b><a NAME="E1:35_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/><b>consider </b><font color="Maroon">m1</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="E3:35_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> holds <br/> <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="tbsp_1.html#D5">TBSP_1:def 5</a></i>;<br/><b>take </b><font color="Maroon">m</font> = <font color="Maroon">m1</font>;<br/><b>thus </b><a NAME="E4:35_1"/>
for <font color="Olive">n</font>, <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> holds <br/> <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></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>
<b>then consider </b><font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">m1</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="E4:35"/><i><font color="Green">E40</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r2</font>
 <b>and </b><br/><a NAME="E5:35"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> holds <br/> <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r2</font>
 ;<br/>
<b>consider </b><font color="Maroon">r1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E7:35"/><i><font color="Green">E44</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font>
 <b>and </b><br/><a NAME="E8:35"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">m2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_6.html#K2" target="_self">dist_to_point</a> <font color="Maroon">S</font>,<span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m2</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="seq_2.html#T16">SEQ_2:16</a></i>;<br/>
<a NAME="E9:35"/><i><font color="Green">E46</font></i>: 
for <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">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font> holds <br/> <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:35_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font>
 ;<br/>

<a NAME="E2:35_2"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="metric_6.html#K2" target="_self">dist_to_point</a> <font color="Maroon">S</font>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:35_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_6.html#K2" target="_self">dist_to_point</a> <font color="Maroon">S</font>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T5">METRIC_1:5</a></i>;<br/>
<a NAME="E4:35_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_6.html#K2" target="_self">dist_to_point</a> <font color="Maroon">S</font>,<span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:35_2"/>
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:35"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> st <br/>( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r2</font> as   <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>take </b>
<font color="Maroon">r</font>
;<br/>

<b>take </b><font color="Maroon">x</font> = <font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m1</font>;<br/>

<a NAME="E2:35_3"/><i><font color="Green">E54</font></i>: 
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r2</font>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E3:35_3"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_3_1_1_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font> or <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:35_3_1_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font>
 ;<br/><div class="add"><a NAME="E2:35_3_1_1_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font>
 <b>by </b><i/>;<br/><a NAME="E3:35_3_1_1_1_1"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E4:35_3_1_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:35_3_1_1_1_1"/>
<font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:35_3_1_1_1_2"/>
<font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:35_3_1_1_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r2</font>
 <b>by </b><i/>;<br/><a NAME="E3:35_3_1_1_1_2"/>
<font color="Maroon">r2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E4:35_3_1_1_1_2"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:35_3_1_1_1_2"/>
<font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:35_3_1"/>
<font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:35_3"/>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:35"/>
<font color="Maroon">S</font> is <a href="metric_6.html#V1" target="_self">bounded</a>
 <b>by </b><i/>;<br/>


</div>
