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

<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/>

<b>assume </b><a NAME="E1:19"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">P</font> is <a href="compts_1.html#V6">compact</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_1"/>
(  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_1"><b>case </b></a><a NAME="E1:19_1_1_1"/><i><font color="Green">E32</font></i>: 
 <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:19_1_1_1"/><i><font color="Green">E33</font></i>: 
 <a href="topmetr.html#K3">R^1</a>  <a href="compts_1.html#NR2" target="_self">is_T2</a> 
 <b>by </b><i><a class="ref" href="pcomps_1.html#T38">PCOMPS_1:38</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/><a NAME="E3:19_1_1_1"/>
for <font color="Olive">s1</font> being  <a href="seq_1.html#NM1">Real_Sequence</a>  st  <a href="relset_1.html#K5">rng</a> <font color="Olive">s1</font> <a href="tarski.html#R1">c=</a>  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">P</font> &amp; <font color="Olive">s1</font> is <a href="seq_2.html#V4">convergent</a> holds <br/> <a href="seq_2.html#K2">lim</a> <font color="Olive">s1</font> <a href="hidden.html#R2">in</a>  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s1</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:19_1_1_1_1"/><i><font color="Green">E34</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a>  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">P</font> &amp; <font color="Maroon">s1</font> is <a href="seq_2.html#V4">convergent</a> )
 ;<br/>

<a NAME="E2:19_1_1_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp; <font color="Maroon">s1</font> is <a href="seq_2.html#V4">convergent</a> )
 
;<br/>
<b>set </b><font color="Maroon">x</font> =  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font>;<br/>
<b>reconsider </b><font color="Maroon">x</font> =  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font> as   <a href="pre_topc.html#NM2">Point</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/>
<b>thus </b><a NAME="E4:19_1_1_1_1"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a>  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1_1_1_1_1"/>
not  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a>  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:19_1_1_1_1_1"/><b>then </b>
not  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
;<br/>
<a NAME="E3:19_1_1_1_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/>
<a NAME="E4:19_1_1_1_1_1"/>
<font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">Otx</font> being   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> , <font color="Maroon">OtP</font> being   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E6:19_1_1_1_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">Otx</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">OtP</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Otx</font> &amp; <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">OtP</font> &amp; <font color="Maroon">Otx</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">OtP</font> )
 <b>by </b><i>, , , <a class="ref" href="compts_1.html#T15">COMPTS_1:15</a></i>;<br/>
<a NAME="E7:19_1_1_1_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">OtP</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E8:19_1_1_1_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Otx</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E9:19_1_1_1_1_1"/><i><font color="Green">E39</font></i>: 
 <a href="topmetr.html#K3">R^1</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <a href="metric_1.html#K8">RealSpace</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K3">Family_open_set</a> <a href="metric_1.html#K8">RealSpace</a> </span>)</span> #)
 
<b>by </b><i><a class="ref" href="pcomps_1.html#D6">PCOMPS_1:def 6</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  ;<br/>
<b>consider </b><font color="Maroon">r</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="E12:19_1_1_1_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Otx</font> )
 <b>by </b><i>, <a class="ref" href="topmetr.html#T22">TOPMETR:22</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</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/>
<a NAME="E14:19_1_1_1_1_1"/><i><font color="Green">E41</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E15:19_1_1_1_1_1"/><i><font color="Green">E52</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a>  :  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Olive">q</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/>
<a NAME="E16:19_1_1_1_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> &amp; not  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:19_1_1_1_1_1_1"/><i><font color="Green">E53</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">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/>

<b>set </b><font color="Maroon">m</font> = <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1;<br/>
<a NAME="E2:19_1_1_1_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E3:19_1_1_1_1_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:19_1_1_1_1_1_1"/><b>then </b>
<a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</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="metric_1.html#D13">METRIC_1:def 13</a></i>;<br/>
<a NAME="E5:19_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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="metric_1.html#T10">METRIC_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> 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="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E7:19_1_1_1_1_1_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#K8">RealSpace</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E8:19_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">y</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="weierstr.html#T1" target="_self">Th1</a>, , <a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E9:19_1_1_1_1_1_1"/>
<font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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_2.html#T6">FUNCT_2:6</a></i>;<br/>
<a NAME="E10:19_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E11:19_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:19_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>


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


</div><b>end;</b></div><b>hence </b><a NAME="E4:19_1_1_1"/>
 <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">P</font> is <a href="rcomp_1.html#V2">closed</a>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D4">RCOMP_1:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_2"><b>case </b></a><a NAME="E1:19_1_1_2"/><i><font color="Green">E61</font></i>: 
 <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:19_1_1_2"/>
for <font color="Olive">s1</font> being  <a href="seq_1.html#NM1">Real_Sequence</a>  st  <a href="relset_1.html#K5">rng</a> <font color="Olive">s1</font> <a href="tarski.html#R1">c=</a>  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">P</font> &amp; <font color="Olive">s1</font> is <a href="seq_2.html#V4">convergent</a> holds <br/> <a href="seq_2.html#K2">lim</a> <font color="Olive">s1</font> <a href="hidden.html#R2">in</a>  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s1</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:19_1_1_2_1"/><i><font color="Green">E62</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a>  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">P</font> &amp; <font color="Maroon">s1</font> is <a href="seq_2.html#V4">convergent</a> )
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:19_1_1_2_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:19_1_1_2_1"/>
<font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</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_2.html#T6">FUNCT_2:6</a></i>;<br/>
<b>hence </b><a NAME="E5:19_1_1_2_1"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a>  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">P</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:19_1_1_2"/>
 <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">P</font> is <a href="rcomp_1.html#V2">closed</a>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D4">RCOMP_1:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:19"/>
 <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">P</font> is <a href="rcomp_1.html#V2">closed</a>
 ;<br/>


</div>
