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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<a NAME="E1:24"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>( <font color="Olive">b<sub>2</sub></font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:24_1"/><i><font color="Green">E12</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 ;<br/>

<a NAME="E2:24_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T22" target="_self">Th22</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E4:24_1"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="seq_2.html#V4">convergent</a> )
 <b>by </b><i><a class="ref" href="seq_4.html#T57">SEQ_4:57</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<a NAME="E5:24_1"/><i><font color="Green">E14</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> is <a href="rcomp_1.html#V2" target="_self">closed</a>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E6:24_1"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a> st <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="rcomp_1.html#E4:24_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/>
<a NAME="E7:24_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T45">RELAT_1:45</a></i>;<br/>
<a NAME="E8:24_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 
<b>by </b><i><a class="txt" href="rcomp_1.html#E1:24_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E9:24_1"/>
( <font color="Maroon">c<sub>4</sub></font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> )
 <b>by </b><i><a class="txt" href="rcomp_1.html#E4:24_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="rcomp_1.html#E5:24_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="rcomp_1.html#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:24"/>
<span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> is <a href="rcomp_1.html#V1" target="_self">compact</a>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D3" target="_self">Def3</a></i>;<br/>


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