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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="E1:29"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">X</font> is <a href="rcomp_1.html#V2">closed</a>
 ;<br/>

<b>let </b><font color="Maroon">s1</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="rcomp_1.html#D4">RCOMP_1:def 4</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:29"/><i><font color="Green">E31</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a>  <a href="pscomp_1.html#K4" target="_self">-</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E3:29"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">s1</font> is <a href="seq_2.html#V4">convergent</a>
 ;<br/>

<a NAME="E4:29"/><i><font color="Green">E34</font></i>: 
 <a href="seq_1.html#K17">-</a> <font color="Maroon">s1</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, <a class="ref" href="seq_2.html#T23">SEQ_2:23</a></i>;<br/>
<a NAME="E5:29"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="seq_1.html#K17">-</a> <font color="Maroon">s1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="seq_1.html#K17">-</a> <font color="Maroon">s1</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:29_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K17">-</a> <font color="Maroon">s1</font></span>)</span>
 <b>and </b><br/><a NAME="E4:29_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_1.html#K17">-</a> <font color="Maroon">s1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E6:29_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="seq_1.html#T14">SEQ_1:14</a></i>;<br/>
<a NAME="E7:29_1"/>
<font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</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/>
<a NAME="E8:29_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K4" target="_self">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" target="_self">-</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pscomp_1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E9:29_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:29"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="seq_1.html#K17">-</a> <font color="Maroon">s1</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#D4">RCOMP_1:def 4</a></i>;<br/>
<a NAME="E7:29"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="seq_1.html#K17">-</a> <font color="Maroon">s1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="seq_2.html#T24">SEQ_2:24</a></i>;<br/>
<a NAME="E8:29"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K4" target="_self">-</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:29"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K4" target="_self">-</a> <font color="Maroon">X</font>
 ;<br/>


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