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

<b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">S1</font> be   <a href="normsp_1.html#NM2">sequence</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>;<br/><b>let </b><font color="Maroon">S</font> be   <a href="normsp_1.html#NM2">sequence</a> of <a href="metric_1.html#K8">RealSpace</a> ;<br/>





<b>assume </b><a NAME="E1:8"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">S</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">S1</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 ;<br/>

<a NAME="E2:8"/><b>then </b><i><font color="Green">E33</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T14">TOPMETR:14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E4:8"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">S</font> is <a href="tbsp_1.html#V2">convergent</a> implies <font color="Maroon">S1</font> is <a href="tbsp_1.html#V2">convergent</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">S</font> is <a href="tbsp_1.html#V2">convergent</a>
 ;<br/>

<b>then consider </b><font color="Maroon">x0</font> being   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a> <b> such that </b><br/><a NAME="E3:8_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>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">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</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>,<font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font>
 <b>by </b><i><a class="ref" href="tbsp_1.html#D3">TBSP_1:def 3</a></i>;<br/>
<a NAME="E4:8_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="tbsp_1.html#K1">lim</a> <font color="Maroon">S</font>
 
<b>by </b><i>, , <a class="ref" href="tbsp_1.html#D4">TBSP_1:def 4</a></i>;<br/>
<a NAME="E5:8_1"/><i><font color="Green">E47</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>  holds  <font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><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/>

<a NAME="E1:8_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:8_1_1"/><b>then </b>
<font color="Maroon">S1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">S1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:8_1_1"/>
<font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:8_1"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="treal_1.html#T4">TREAL_1:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x0</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <b>by </b><i>, , , , , <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<a NAME="E8:8_1"/>
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>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">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span>,<font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:8_1_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon">n0</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="E3:8_1_2"/><i><font color="Green">E61</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="Maroon">n0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</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>,<font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:8_1_2"/>
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="Maroon">n0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span>,<font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1"><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="E1:8_1_2_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">n0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">x1</font> = 
the <a href="metric_1.html#U1">distance</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="metric_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">x1</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
<br/>.= 
the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#K8">RealSpace</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">x1</font>
<b>by </b><i><a class="ref" href="topmetr.html#D1">TOPMETR:def 1</a></i>
<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="Maroon">m</font></span>)</span>,<font color="Maroon">x0</font>
<b>by </b><i>, <a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E3:8_1_2_1"/>
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:8_1_2"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>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">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span>,<font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:8_1"/>
<font color="Maroon">S1</font> is <a href="tbsp_1.html#V2">convergent</a>
 <b>by </b><i><a class="ref" href="tbsp_1.html#D3">TBSP_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:8"/>
( <font color="Maroon">S1</font> is <a href="tbsp_1.html#V2">convergent</a> implies <font color="Maroon">S</font> is <a href="tbsp_1.html#V2">convergent</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_2"/>
<font color="Maroon">S1</font> is <a href="tbsp_1.html#V2">convergent</a>
 ;<br/>

<b>then consider </b><font color="Maroon">x0</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span><b> such that </b><br/><a NAME="E3:8_2"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>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">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span>,<font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font>
 <b>by </b><i><a class="ref" href="tbsp_1.html#D3">TBSP_1:def 3</a></i>;<br/>
<a NAME="E4:8_2"/>
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x0</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E6:8_2"/>
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>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">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</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>,<font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:8_2_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">n0</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="E3:8_2_1"/><i><font color="Green">E71</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="Maroon">n0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span>,<font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:8_2_1"/>
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="Maroon">n0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</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>,<font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1_1"><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="E1:8_2_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">n0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">x0</font> = 
the <a href="metric_1.html#U1">distance</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="metric_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">x0</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
<br/>.= 
the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#K8">RealSpace</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">x0</font>
<b>by </b><i><a class="ref" href="topmetr.html#D1">TOPMETR:def 1</a></i>
<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="Maroon">m</font></span>)</span>,<font color="Maroon">x1</font>
<b>by </b><i>, <a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E3:8_2_1_1"/>
 <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>,<font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:8_2_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>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">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</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>,<font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:8_2"/>
<font color="Maroon">S</font> is <a href="tbsp_1.html#V2">convergent</a>
 <b>by </b><i><a class="ref" href="tbsp_1.html#D3">TBSP_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:8"/>
( <font color="Maroon">S</font> is <a href="tbsp_1.html#V2">convergent</a> iff <font color="Maroon">S1</font> is <a href="tbsp_1.html#V2">convergent</a> )
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E7:8"/>
( <font color="Maroon">S</font> is <a href="tbsp_1.html#V2">convergent</a> implies  <a href="tbsp_1.html#K1">lim</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="tbsp_1.html#K1">lim</a> <font color="Maroon">S1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_3"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">S</font> is <a href="tbsp_1.html#V2">convergent</a>
 ;<br/>

<a NAME="E2:8_3"/>
 <a href="tbsp_1.html#K1">lim</a> <font color="Maroon">S1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>
 
;<br/>
<b>then reconsider </b><font color="Maroon">t0</font> =  <a href="tbsp_1.html#K1">lim</a> <font color="Maroon">S1</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i>, <a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E4:8_3"/>
for <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r1</font> holds <br/> ex <font color="Olive">m1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">n1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n1</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">n1</font></span>)</span>,<font color="Maroon">t0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r1</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:8_3_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font>
 ;<br/>

<b>then 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="E3:8_3_1"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">n1</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">n1</font> holds <br/> <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n1</font></span>)</span>,<span class="p1">(<span class="default"><a href="tbsp_1.html#K1">lim</a> <font color="Maroon">S1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font>
 <b>by </b><i>, , <a class="ref" href="tbsp_1.html#D4">TBSP_1:def 4</a></i>;<br/>
<a NAME="E4:8_3_1"/>
for <font color="Olive">n1</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">n1</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">n1</font></span>)</span>,<font color="Maroon">t0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:8_3_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font>
 ;<br/>

 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n1</font></span>)</span>,<span class="p1">(<span class="default"><a href="tbsp_1.html#K1">lim</a> <font color="Maroon">S1</font></span>)</span> = 
the <a href="metric_1.html#U1">distance</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="metric_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n1</font></span>)</span>,<span class="p1">(<span class="default"><a href="tbsp_1.html#K1">lim</a> <font color="Maroon">S1</font></span>)</span>
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
<br/>.= 
the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#K8">RealSpace</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n1</font></span>)</span>,<span class="p1">(<span class="default"><a href="tbsp_1.html#K1">lim</a> <font color="Maroon">S1</font></span>)</span>
<b>by </b><i><a class="ref" href="topmetr.html#D1">TOPMETR:def 1</a></i>
<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="Maroon">n1</font></span>)</span>,<font color="Maroon">t0</font>
<b>by </b><i>, <a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E3:8_3_1_1"/>
 <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">n1</font></span>)</span>,<font color="Maroon">t0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:8_3_1"/>
 ex <font color="Olive">m1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">n1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n1</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">n1</font></span>)</span>,<font color="Maroon">t0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:8_3"/>
 <a href="tbsp_1.html#K1">lim</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="tbsp_1.html#K1">lim</a> <font color="Maroon">S1</font>
 <b>by </b><i>, <a class="ref" href="tbsp_1.html#D4">TBSP_1:def 4</a></i>;<br/>


</div><b>end;</b></div>


</div>
