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<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="E1:20"/><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="20_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_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="20_1_1_1"><b>case </b></a><a NAME="E1:20_1_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> 
 ;<br/><div class="add"><a NAME="E2:20_1_1_1"/><i><font color="Green">E32</font></i>: 
 <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">P</font> is <a href="seq_4.html#V3">bounded</a>
 <b>by </b><i>, </i>;<br/><a NAME="E3:20_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>, </i>;<br/><b>hence </b><a NAME="E4:20_1_1_1"/>
 <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">P</font> is <a href="rcomp_1.html#V1">compact</a>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T29">RCOMP_1:29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2"><b>case </b></a><a NAME="E1:20_1_1_2"/><i><font color="Green">E33</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"><b>assume </b><a NAME="E2:20_1_1_2"/>
not  <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">P</font> is <a href="rcomp_1.html#V1">compact</a>
 ;<br/><b>then consider </b><font color="Maroon">s1</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E4:20_1_1_2"/><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; ( for <font color="Olive">s2</font> being  <a href="seq_1.html#NM1">Real_Sequence</a>  holds <br/>(  not <font color="Olive">s2</font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">s1</font> or  not <font color="Olive">s2</font> is <a href="seq_2.html#V4">convergent</a> or  not  <a href="seq_2.html#K2">lim</a> <font color="Olive">s2</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><a class="ref" href="rcomp_1.html#D3">RCOMP_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:20_1_1_2"/><i><font color="Green">E35</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="E7:20_1_1_2"/>
<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="E8:20_1_1_2"/>
 <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">P</font> is <a href="rcomp_1.html#V1">compact</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:20"/>
 <a href="weierstr.html#K3" target="_self">[#]</a> <font color="Maroon">P</font> is <a href="rcomp_1.html#V1">compact</a>
 ;<br/>


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