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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> is <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a> implies <font color="Maroon" title="c1">X</font> is <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> )</span><br/>

<b>assume </b><a NAME="E1:14"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a></span><br/>

<b>assume </b><a NAME="E2:14"/>
not <font color="Maroon" title="c1">X</font> is <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c2">s1</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><b> such that </b><br/><a NAME="E4:14"/><i><font color="Green" title="E10">A2</font></i>: 
(  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c2">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; not  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="ref" href="rcomp_1.html#D4" target="_self" title="RCOMP_1:def.4">Def4</a></i>;<br/>
<a NAME="E5:14"/>
 ex <font color="Olive" title="b1">s2</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> st <br/>( <font color="Olive" title="b1">s2</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Maroon" title="c2">s1</font> &amp; <font color="Olive" title="b1">s2</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Olive" title="b1">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="txt" href="#E1:14"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="#E4:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="rcomp_1.html#D3" target="_self" title="RCOMP_1:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E6:14"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="seq_4.html#T30" title="SEQ_4:th.30">SEQ_4:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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