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

<b>let </b><font color="Maroon">X</font> be  <a href="setfam_1.html#V1">without_zero</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:49"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">X</font> is <a href="rcomp_1.html#V2">closed</a>
 <b>and </b><br/><a NAME="E2:49"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">X</font> is <a href="seq_4.html#V3">bounded</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="E3:49"/><i><font color="Green">E32</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#K5" target="_self">Inv</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E4:49"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">s1</font> is <a href="seq_2.html#V4">convergent</a>
 ;<br/>

<a NAME="E5:49"/><i><font color="Green">E35</font></i>: 
not 0 <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="txt" href="pscomp_1.html#E3:49"><i><font color="Green">E32</font></i></a>, </i>;<br/>
<div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1"/>
not <font color="Maroon">s1</font> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 ;<br/><a NAME="E2:49_1"/><b>then </b>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/><b>hence </b><a NAME="E3:49_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pscomp_1.html#E4:49"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_2"/><i><font color="Green">E41</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:49_2"/><i><font color="Green">E43</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:49_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K18">"</a> </span>)</span>
 ;<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="E3:49_2_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K18">"</a> </span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</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">x</font> = <font color="Maroon">x</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="E5:49_2_1"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K2">"</a> 
 
<b>by </b><i><a class="ref" href="seq_1.html#D8">SEQ_1:def 8</a></i>;<br/>
<a NAME="E6:49_2_1"/>
<font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</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="E7:49_2_1"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K5" target="_self">Inv</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" target="_self">Inv</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E3:49"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E8:49_2_1"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:49_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="txt" href="pscomp_1.html#E7:49"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:49_2"/>
not <font color="Maroon">s1</font> <a href="seq_1.html#K18">"</a>  is <a href="seq_2.html#V3">bounded</a>
 <b>by </b><i>, <a class="txt" href="pscomp_1.html#E5:49"><i><font color="Green">E35</font></i></a>, , </i>;<br/><a NAME="E4:49_2"/><b>then </b>
not  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K18">"</a> </span>)</span> is <a href="seq_4.html#V3">bounded</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:49_2"/>
contradiction
 <b>by </b><i><a class="txt" href="pscomp_1.html#E2:49"><i><font color="Green">E31</font></i></a>, , <a class="ref" href="rcomp_1.html#T5">RCOMP_1:5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:49"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">s1</font> <a href="seq_1.html#K18">"</a>  is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, <a class="txt" href="pscomp_1.html#E5:49"><i><font color="Green">E35</font></i></a>, <a class="ref" href="seq_2.html#T35">SEQ_2:35</a></i>;<br/>
<a NAME="E9:49"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3"><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="E1:49_3"/>
<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"><font color="Maroon">s1</font> <a href="seq_1.html#K18">"</a> </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="E3:49_3"/><i><font color="Green">E49</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"><font color="Maroon">s1</font> <a href="seq_1.html#K18">"</a> </span>)</span>
 <b>and </b><br/><a NAME="E4:49_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K18">"</a> </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="E6:49_3"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</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> <a href="real_1.html#K2">"</a> 
 
<b>by </b><i>, <a class="ref" href="seq_1.html#D8">SEQ_1:def 8</a></i>;<br/>
<a NAME="E7:49_3"/>
<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="E8:49_3"/><b>then </b>
1 <a href="real_1.html#K6">/</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> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K5" target="_self">Inv</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" target="_self">Inv</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E3:49"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E9:49_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K5" target="_self">Inv</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" target="_self">Inv</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E7:49"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>;<br/>
<b>hence </b><a NAME="E10:49_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E10:49"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E1:49"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="rcomp_1.html#D4">RCOMP_1:def 4</a></i>;<br/>
<a NAME="E11:49"/>
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="real_1.html#K2">"</a>  <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K18">"</a> </span>)</span>
 
<b>by </b><i>, <a class="txt" href="pscomp_1.html#E5:49"><i><font color="Green">E35</font></i></a>, <a class="txt" href="pscomp_1.html#E6:49"><i><font color="Green">E36</font></i></a>, <a class="ref" href="seq_2.html#T36">SEQ_2:36</a></i>;<br/>
<a NAME="E12:49"/><b>then </b>
1 <a href="real_1.html#K6">/</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#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>;<br/>
<a NAME="E13:49"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</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#K5" target="_self">Inv</a> <font color="Maroon">X</font>
 
;<br/>
<b>hence </b><a NAME="E14:49"/>
 <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#K5" target="_self">Inv</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T56">XCMPLX_1:56</a></i>;<br/>


</div>
