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

<b>let </b><font color="Maroon" title="c1">seq</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c2">Eseq</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ;<br/>



<b>assume </b><a NAME="E1:17"/><i><font color="Green" title="E14">A1</font></i>: 
( <font color="Maroon" title="c1">seq</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">Eseq</font> &amp; <font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> )
 ;<br/>

<a NAME="E2:17"/><i><font color="Green" title="E15">A2</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">Eseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">s</font> =  <a href="rinfsup1.html#K2" title="RINFSUP1:func.2">inf</a> <font color="Maroon" title="c1">seq</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E4:17"/>
for <font color="Olive" title="b1">x</font> being <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">Eseq</font> holds <br/><font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:17_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">Eseq</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">n1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:17_1"/><i><font color="Green" title="E16">A3</font></i>: 
( <font color="Maroon" title="c5">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c2">Eseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> )
 <b>by </b><i><a class="txt" href="prob_4.html#E2:17"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>thus </b><a NAME="E4:17_1"/>
<font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="prob_4.html#E1:17"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="prob_4.html#E3:17_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="rinfsup1.html#T8" title="RINFSUP1:th.8">RINFSUP1:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:17"/><b>then </b><i><font color="Green" title="E16">A4</font></i>: 
<font color="Maroon" title="c3">s</font> is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">Eseq</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D10" title="SUPINF_1:def.10">SUPINF_1:def 10</a></i>;<br/>
<a NAME="E6:17"/><b>then </b><i><font color="Green" title="E17">A5</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">Eseq</font> is <a href="supinf_1.html#V4" title="SUPINF_1:attr.4">bounded_below</a>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D12" title="SUPINF_1:def.12">SUPINF_1:def 12</a></i>;<br/>
<a NAME="E7:17"/><i><font color="Green" title="E18">A6</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">Eseq</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">Eseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">k1</font> =  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">Eseq</font> <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">n1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:17_2"/><i><font color="Green" title="E19">A7</font></i>: 
( <font color="Maroon" title="c5">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c2">Eseq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k1</font> )
 <b>by </b><i><a class="txt" href="prob_4.html#E2:17"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">n1</font> = <font color="Maroon" title="c5">n1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="prob_4.html#E4:17_2"><i><font color="Green" title="E19">A7</font></i></a></i>;<br/>
<a NAME="E6:17_2"/>
<font color="Maroon" title="c1">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k1</font>
 
<b>by </b><i><a class="txt" href="prob_4.html#E1:17"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="prob_4.html#E4:17_2"><i><font color="Green" title="E19">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:17_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:17"/><b>then </b><i><font color="Green" title="E19">A8</font></i>: 
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">Eseq</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 
<b>by </b><i><a class="txt" href="prob_4.html#E6:17"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="supinf_1.html#T84" title="SUPINF_1:th.84">SUPINF_1:84</a></i>;<br/>
<a NAME="E9:17"/><i><font color="Green" title="E20">A9</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">Eseq</font></span>)</span>
 
<b>by </b><i><a class="txt" href="prob_4.html#E5:17"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="supinf_1.html#D17" title="SUPINF_1:def.17">SUPINF_1:def 17</a></i>;<br/>
<a NAME="E10:17"/>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">Eseq</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3"><b>proof </b></a><div class="add">

<a NAME="E1:17_3"/><i><font color="Green" title="E21">A10</font></i>: 
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">Eseq</font></span>)</span> is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">Eseq</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D17" title="SUPINF_1:def.17">SUPINF_1:def 17</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">r1</font> =  <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">Eseq</font></span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="prob_4.html#E6:17"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="prob_4.html#E7:17"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="supinf_1.html#T84" title="SUPINF_1:th.84">SUPINF_1:84</a></i>;<br/>
<a NAME="E3:17_3"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c4">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:17_3_1"/><i><font color="Green" title="E22">A11</font></i>: 
<font color="Maroon" title="c1">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Eseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font>
 
<b>by </b><i><a class="txt" href="prob_4.html#E1:17"><i><font color="Green" title="E14">A1</font></i></a></i>;<br/>
<a NAME="E2:17_3_1"/>
<font color="Maroon" title="c2">Eseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">Eseq</font>
 
<b>by </b><i><a class="txt" href="prob_4.html#E2:17"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E3:17_3_1"/><b>then </b>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">Eseq</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">Eseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font>
 
<b>by </b><i><a class="txt" href="prob_4.html#E1:17_3"><i><font color="Green" title="E21">A10</font></i></a>, <a class="ref" href="supinf_1.html#D10" title="SUPINF_1:def.10">SUPINF_1:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">r2</font>, <font color="Maroon" title="c7">r3</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:17_3_1"/><i><font color="Green" title="E23">A12</font></i>: 
( <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">Eseq</font></span>)</span> &amp; <font color="Maroon" title="c7">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Eseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c6">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">r3</font> )
 <b>by </b><i><a class="txt" href="prob_4.html#E8:17"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="prob_4.html#E1:17_3_1"><i><font color="Green" title="E22">A11</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:17_3_1"/>
<font color="Maroon" title="c4">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="prob_4.html#E1:17"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="prob_4.html#E5:17_3_1"><i><font color="Green" title="E23">A12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:17_3"/>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">Eseq</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="ref" href="rinfsup1.html#T10" title="RINFSUP1:th.10">RINFSUP1:10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:17"/>
 <a href="rinfsup1.html#K2" title="RINFSUP1:func.2">inf</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">Eseq</font></span>)</span>
 <b>by </b><i><a class="txt" href="prob_4.html#E9:17"><i><font color="Green" title="E20">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


</div>
