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<b>let </b><font color="Maroon">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">g</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<a NAME="E1:23"/>
for <font color="Olive">s1</font> being  <a href="seq_1.html#NM1">Real_Sequence</a>  st  <a href="relat_1.html#K2">rng</a> <font color="Olive">s1</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">s</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> &amp; <font color="Olive">s1</font> is <a href="seq_2.html#V4">convergent</a> holds <br/> <a href="seq_2.html#K2">lim</a> <font color="Olive">s1</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">s</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s1</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:23_1"/><i><font color="Green">E41</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">s1</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">s</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 <b>and </b><br/><a NAME="E2:23_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">s1</font> is <a href="seq_2.html#V4">convergent</a>
 ;<br/>

<a NAME="E3:23_1"/><i><font color="Green">E44</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:23_1_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">s2</font> = <a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">g</font> as   <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E3:23_1_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E4:23_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:23_1_1"/><i><font color="Green">E52</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="seq_4.html#T40">SEQ_4:40</a></i>;<br/>
<a NAME="E6:23_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">s2</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T39">SEQ_4:39</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:23_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><a NAME="E2:23_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:23_1_1_1"/><b>then </b>
<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="relat_1.html#K2">rng</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:23_1_1_1"/><b>then </b>
<font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</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">s</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 <b>by </b><i/>;<br/><a NAME="E5:23_1_1_1"/><b>then </b>
<font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> ) </span>}</span> 
 ;<br/><a NAME="E6:23_1_1_1"/><b>then </b>
 ex <font color="Olive">p</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> )
 ;<br/><b>hence </b><a NAME="E7:23_1_1_1"/>
<font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:23_1_1"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font>
 <b>by </b><i>, , , <a class="ref" href="seq_2.html#T32">SEQ_2:32</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:23_1"/>
<font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:23_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">s2</font> = <a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">s</font> as   <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E3:23_1_2"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E4:23_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:23_1_2"/><i><font color="Green">E56</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="seq_4.html#T40">SEQ_4:40</a></i>;<br/>
<a NAME="E6:23_1_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">s2</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T39">SEQ_4:39</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:23_1_2_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><a NAME="E2:23_1_2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:23_1_2_1"/><b>then </b>
<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="relat_1.html#K2">rng</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:23_1_2_1"/><b>then </b>
<font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</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">s</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 <b>by </b><i/>;<br/><a NAME="E5:23_1_2_1"/><b>then </b>
<font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> ) </span>}</span> 
 ;<br/><a NAME="E6:23_1_2_1"/><b>then </b>
 ex <font color="Olive">p</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> )
 ;<br/><b>hence </b><a NAME="E7:23_1_2_1"/>
<font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:23_1_2"/>
<font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font>
 <b>by </b><i>, , , <a class="ref" href="seq_2.html#T32">SEQ_2:32</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:23_1"/><b>then </b>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:23_1"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</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">s</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:23"/>
<span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">g</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/>;<br/>


</div>
