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<b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:27"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">X</font> is <a href="rcomp_1.html#V1" target="_self">compact</a>
 ;<br/>

<b>assume </b><a NAME="E2:27"/>
not <font color="Maroon">X</font> is <a href="rcomp_1.html#V2" target="_self">closed</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="E4:27"/><i><font color="Green">E42</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">s1</font> is <a href="seq_2.html#V4">convergent</a> &amp; not  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/>
<a NAME="E5:27"/>
 ex <font color="Olive">s2</font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>( <font color="Olive">s2</font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">s1</font> &amp; <font color="Olive">s2</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Olive">s2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E6:27"/>
contradiction
 <b>by </b><i>, <a class="ref" href="seq_4.html#T30">SEQ_4:30</a></i>;<br/>


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