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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be  <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a> <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> ;<br/><b>let </b><font color="Maroon" title="c3">s1</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="rcomp_1.html#D3" title="RCOMP_1:def.3">RCOMP_1:def 3</a><br/>



<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E7">A1</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">B</font>
 ;<br/>

<a NAME="E2:9"/>
<font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E3:9"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E1:9"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">s2</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><b> such that </b><br/><a NAME="E5:9"/><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c4">s2</font> is    <a href="seqm_3.html#M1" title="SEQM_3:mode.1">subsequence</a> of <font color="Maroon" title="c3">s1</font>
 <b>and </b><br/><a NAME="E6:9"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c4">s2</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>and </b><br/><a NAME="E7:9"/><i><font color="Green" title="E10">A4</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D3" title="RCOMP_1:def.3">RCOMP_1:def 3</a></i>;<br/>
<a NAME="E8:9"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">s2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">s1</font>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E5:9"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="rfunct_2.html#T11" title="RFUNCT_2:th.11">RFUNCT_2:11</a></i>;<br/>
<a NAME="E9:9"/><b>then </b><i><font color="Green" title="E11">A5</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">s2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">B</font>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E1:9"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E10:9"/>
<font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E11:9"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">s2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E9:9"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">s3</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><b> such that </b><br/><a NAME="E13:9"/><i><font color="Green" title="E12">A6</font></i>: 
<font color="Maroon" title="c5">s3</font> is    <a href="seqm_3.html#M1" title="SEQM_3:mode.1">subsequence</a> of <font color="Maroon" title="c4">s2</font>
 <b>and </b><br/><a NAME="E14:9"/><i><font color="Green" title="E13">A7</font></i>: 
<font color="Maroon" title="c5">s3</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>and </b><br/><a NAME="E15:9"/><i><font color="Green" title="E14">A8</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c5">s3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D3" title="RCOMP_1:def.3">RCOMP_1:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">s3</font>
;<br/>

<b>thus </b><a NAME="E16:9"/>
<font color="Maroon" title="c5">s3</font> is    <a href="seqm_3.html#M1" title="SEQM_3:mode.1">subsequence</a> of <font color="Maroon" title="c3">s1</font>
 <b>by </b><i><a class="txt" href="jct_misc.html#E5:9"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="jct_misc.html#E13:9"><i><font color="Green" title="E12">A6</font></i></a>, <a class="ref" href="seqm_3.html#T48" title="SEQM_3:th.48">SEQM_3:48</a></i>;<br/>

<b>thus </b><a NAME="E17:9"/>
<font color="Maroon" title="c5">s3</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="jct_misc.html#E14:9"><i><font color="Green" title="E13">A7</font></i></a></i>;<br/>

<a NAME="E18:9"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c5">s3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s2</font>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E6:9"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="jct_misc.html#E13:9"><i><font color="Green" title="E12">A6</font></i></a>, <a class="ref" href="seq_4.html#T30" title="SEQ_4:th.30">SEQ_4:30</a></i>;<br/>
<b>hence </b><a NAME="E19:9"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c5">s3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="jct_misc.html#E7:9"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="jct_misc.html#E15:9"><i><font color="Green" title="E14">A8</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


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