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

<b>thus </b><a NAME="E1:84"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a> implies for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <font color="Olive">r</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:84_1"/>
<font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a>
 ;<br/>

<a NAME="E2:84_1"/><b>then </b><i><font color="Green">E33</font></i>: 
( <font color="Maroon">seq</font> is <a href="seq_2.html#V1">bounded_above</a> &amp; <font color="Maroon">seq</font> is <a href="seq_2.html#V2">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a></i>;<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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_1_1"/>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> or 0 <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> or <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_1"><b>suppose </b></a><a NAME="E1:84_1_1_1"/><i><font color="Green">E34</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E2:84_1_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V1">bounded_above</a>
 <b>by </b><i>, <a class="txt" href="seqm_3.html#E11"><i><font color="Green">Lemma49</font></i></a></i>;<br/><a NAME="E3:84_1_1_1"/>
<font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V2">bounded_below</a>
 <b>by </b><i>, , <a class="txt" href="seqm_3.html#E12"><i><font color="Green">Lemma50</font></i></a></i>;<br/><b>hence </b><a NAME="E4:84_1_1_1"/>
<font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#D7" target="_self">Def7</a>, <a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_2"><b>suppose </b></a><a NAME="E1:84_1_1_2"/>
0 <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:84_1_1_2"/>
<font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a>
 <b>by </b><i><a class="txt" href="seqm_3.html#E11"><i><font color="Green">Lemma49</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_3"><b>suppose </b></a><a NAME="E1:84_1_1_3"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:84_1_1_3"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V1">bounded_above</a>
 <b>by </b><i>, <a class="txt" href="seqm_3.html#E12"><i><font color="Green">Lemma50</font></i></a></i>;<br/><a NAME="E3:84_1_1_3"/>
<font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V2">bounded_below</a>
 <b>by </b><i>, , <a class="txt" href="seqm_3.html#E11"><i><font color="Green">Lemma49</font></i></a></i>;<br/><b>hence </b><a NAME="E4:84_1_1_3"/>
<font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#D7" target="_self">Def7</a>, <a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:84_2"/>
<font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a>
 ;<br/><a NAME="E2:84_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#D6" target="_self">Def6</a></i>;<br/><b>hence </b><a NAME="E3:84_2"/>
 <a href="seq_1.html#K17">-</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a>
 <b>by </b><i><a class="ref" href="seq_1.html#T25">SEQ_1:25</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E3:84"/>
( <font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a> implies  <a href="seq_1.html#K22">abs</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:84_3"/>
<font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a>
 ;<br/>

<b>then consider </b><font color="Maroon">r</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="E3:84_3"/><i><font color="Green">E42</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E4:84_3"/><i><font color="Green">E64</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  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="seq_2.html#T15">SEQ_2:15</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_3_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:84_3_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/><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="E2:84_3_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:84_3_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:84_3"/>
 <a href="seq_1.html#K22">abs</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a>
 <b>by </b><i><a class="ref" href="seq_2.html#T15">SEQ_2:15</a></i>;<br/>


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

<b>assume </b><a NAME="E4:84"/>
 <a href="seq_1.html#K22">abs</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a>
 ;<br/>

<b>then consider </b><font color="Maroon">r</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="E6:84"/><i><font color="Green">E65</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> ) )
 <b>by </b><i><a class="ref" href="seq_2.html#T15">SEQ_2:15</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_4"><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 href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>

;<br/><b>hence </b><a NAME="E2:84_4"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <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="E8:84"/>
<font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a>
 <b>by </b><i>, <a class="ref" href="seq_2.html#T15">SEQ_2:15</a></i>;<br/>


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