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

<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="E1:30"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">X</font> is <a href="seq_4.html#V3">bounded</a> &amp; <font color="Maroon">X</font> is <a href="rcomp_1.html#V2" target="_self">closed</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </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><a NAME="E1:30_1"/><i><font color="Green">E42</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:30_1"/>
<font color="Maroon">s1</font> is <a href="seq_2.html#V3">bounded</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:30_1_1"/>
<font color="Maroon">s1</font> is <a href="seq_2.html#V1">bounded_above</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:30_1_1_1"/>
<font color="Maroon">X</font> is <a href="seq_4.html#V1">bounded_above</a>
 
<b>by </b><i>, <a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">p</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="E3:30_1_1_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">q</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="seq_4.html#D1">SEQ_4:def 1</a></i>;<br/>
<b>take </b><font color="Maroon">r</font> = <font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> 1; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_2.html#D3">SEQ_2:def 3</a><br/>

<a NAME="E4:30_1_1_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">t</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s1</font> holds <br/><font color="Olive">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</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="E1:30_1_1_1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s1</font>
 ;<br/>

<a NAME="E2:30_1_1_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:30_1_1_1_1"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<b>hence </b><a NAME="E4:30_1_1_1_1"/>
<font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</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="E5:30_1_1_1"/>
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="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_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="E1:30_1_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<a NAME="E2:30_1_1_1_2"/><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="E3:30_1_1_1_2"/><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/>
<b>hence </b><a NAME="E4:30_1_1_1_2"/>
<font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:30_1_1_1"/>
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:30_1_1"/>
<font color="Maroon">s1</font> is <a href="seq_2.html#V2">bounded_below</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:30_1_1_2"/>
<font color="Maroon">X</font> is <a href="seq_4.html#V2">bounded_below</a>
 
<b>by </b><i>, <a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">p</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="E3:30_1_1_2"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">q</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font>
 <b>by </b><i><a class="ref" href="seq_4.html#D2">SEQ_4:def 2</a></i>;<br/>
<b>take </b><font color="Maroon">r</font> = <font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> 1; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_2.html#D4">SEQ_2:def 4</a><br/>

<a NAME="E4:30_1_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> 1</span>)</span>
 
;<br/>
<a NAME="E5:30_1_1_2"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">t</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s1</font> holds <br/><font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">t</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</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="E1:30_1_1_2_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s1</font>
 ;<br/>

<a NAME="E2:30_1_1_2_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:30_1_1_2_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<b>hence </b><a NAME="E4:30_1_1_2_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t</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="E6:30_1_1_2"/>
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">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2_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="E1:30_1_1_2_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<a NAME="E2:30_1_1_2_2"/><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="E3:30_1_1_2_2"/><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/>
<b>hence </b><a NAME="E4:30_1_1_2_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&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="E7:30_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">s2</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E4:30_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">s2</font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">s1</font> &amp; <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#T57">SEQ_4:57</a></i>;<br/><a NAME="E5:30_1"/>
 ex <font color="Olive">Nseq</font> being <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a> st <font color="Maroon">s2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">s1</font> <a href="seqm_3.html#K3">*</a> <font color="Olive">Nseq</font>
 <b>by </b><i>, <a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/><a NAME="E6:30_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">s2</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T45">RELAT_1:45</a></i>;<br/><a NAME="E7:30_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">s2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:30_1"/><b>then </b>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E9:30_1"/>
 ex <font color="Olive">s2</font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>( <font color="Olive">s2</font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">s1</font> &amp; <font color="Olive">s2</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Olive">s2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:30"/>
<font color="Maroon">X</font> is <a href="rcomp_1.html#V1" target="_self">compact</a>
 <b>by </b><i/>;<br/>


</div>
