<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">X</font> be  <a href="rcomp_1.html#V3" target="_self">open</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:40"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E2:40"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">N</font> being   <a href="rcomp_1.html#M1" target="_self">Neighbourhood</a> of <font color="Maroon">r</font>  holds not <font color="Olive">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 ;<br/>

<div><i><font color="Green">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">N</font> be    <a href="rcomp_1.html#M1" target="_self">Neighbourhood</a> of <font color="Maroon">r</font>;<br/><a NAME="E1:40_1"/>
not <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>consider </b><font color="Maroon">g</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E5:40_1"/><i><font color="Green">E51</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> &amp; <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2" target="_self">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2" target="_self">.[</a></span> )
 <b>by </b><i/>;<br/><a NAME="E6:40_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E8:40_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font> )
 ;<br/><b>take </b><font color="Maroon">s</font> = <font color="Maroon">s</font>;<br/><b>thus </b><a NAME="E9:40_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i>, </i>;<br/><a NAME="E10:40_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E11:40_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> 
 ;<br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ] <b>means </b>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2" target="_self">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  );<br/>
<a NAME="E4:40"/><i><font color="Green">E53</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>   ex <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">s</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </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="E1:40_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/>
<a NAME="E2:40_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">N</font> = <span class="p1"><a href="rcomp_1.html#K2" target="_self">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2" target="_self">.[</a></span> as    <a href="rcomp_1.html#M1" target="_self">Neighbourhood</a> of <font color="Maroon">r</font> <b>by </b><i/>;<br/>
<a NAME="E4:40_2"/>
 ex <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:40_2"/>
 ex <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Olive">s</font>]
 ;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">s1</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E6:40"/><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">s1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">s2</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E8:40"/><i><font color="Green">E55</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">H<sub>2</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon">s3</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E10:40"/><i><font color="Green">E56</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">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="rcomp_1.html#S1" target="_self">RCOMP_1:sch 1</a>();<br/></i>
<div><i><font color="Green">E66</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3"><b>now </b></a><div class="add"><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>assume </b><a NAME="E1:40_3"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 ;<br/><a NAME="E2:40_3"/><b>then </b><i><font color="Green">E68</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K5">"</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/><b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:40_3"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">s</font> <a href="xcmplx_0.html#K5">"</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="seq_4.html#T10">SEQ_4:10</a></i>;<br/><a NAME="E5:40_3"/>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E6:40_3"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T78">XREAL_1:78</a></i>;<br/><a NAME="E7:40_3"/><b>then </b><i><font color="Green">E70</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T218">XCMPLX_1:218</a></i>;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E8:40_3"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E9:40_3"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E10:40_3"/><b>then </b><i><font color="Green">E71</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T120">XREAL_1:120</a></i>;<br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>

<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><b>hence </b><a NAME="E12:40_3"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:40"/><b>then </b><i><font color="Green">E72</font></i>: 
<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#D6">SEQ_2:def 6</a></i>;<br/>
<a NAME="E13:40"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<div><i><font color="Green">E78</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_4"><b>now </b></a><div class="add"><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>assume </b><a NAME="E1:40_4"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 ;<br/><a NAME="E2:40_4"/><b>then </b><i><font color="Green">E79</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K5">"</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/><b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:40_4"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">s</font> <a href="xcmplx_0.html#K5">"</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="seq_4.html#T10">SEQ_4:10</a></i>;<br/><a NAME="E5:40_4"/>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E6:40_4"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T78">XREAL_1:78</a></i>;<br/><a NAME="E7:40_4"/><b>then </b><i><font color="Green">E81</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T218">XCMPLX_1:218</a></i>;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E8:40_4"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E9:40_4"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E10:40_4"/><b>then </b><i><font color="Green">E82</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T120">XREAL_1:120</a></i>;<br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i/>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><b>hence </b><a NAME="E12:40_4"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:40"/><b>then </b><i><font color="Green">E83</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_2.html#D6">SEQ_2:def 6</a></i>;<br/>
<a NAME="E16:40"/><b>then </b><i><font color="Green">E84</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<a NAME="E17:40"/><i><font color="Green">E85</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 <br/> ( <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5"><b>proof </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="E1:40_5"/>
<font color="Maroon">s3</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#K2" target="_self">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2" target="_self">.[</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:40_5"/><b>then </b>
<font color="Maroon">s3</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">s</font> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) </span>}</span> 
 
;<br/>
<a NAME="E3:40_5"/><b>then </b><i><font color="Green">E99</font></i>: 
 ex <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E4:40_5"/>
<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">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E5:40_5"/>
<font color="Maroon">s3</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>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:40"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">s3</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, , , , <a class="ref" href="seq_2.html#T33">SEQ_2:33</a></i>;<br/>
<a NAME="E19:40"/><i><font color="Green">E101</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 
<b>by </b><i>, , , , , <a class="ref" href="seq_2.html#T34">SEQ_2:34</a></i>;<br/>
<a NAME="E20:40"/><i><font color="Green">E102</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">s3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_6"><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="E1:40_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s3</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_6"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s3</font>
 <b>and </b><br/><a NAME="E4:40_6"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</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">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:40_6"/>
<font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:40_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E21:40"/>
<font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  is <a href="rcomp_1.html#V2" target="_self">closed</a>
 
<b>by </b><i/>;<br/>
<a NAME="E22:40"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E23:40"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <font color="Maroon">X</font>
 
;<br/>
<b>hence </b><a NAME="E24:40"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div>
