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<b>let </b><font color="Maroon">s</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>

<b>thus </b><a NAME="E1:13"/>
( <font color="Maroon">s</font> is <a href="comseq_2.html#V1" target="_self">bounded</a> implies  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">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 <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1"/>
<font color="Maroon">s</font> is <a href="comseq_2.html#V1" target="_self">bounded</a>
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:13_1"/><i><font color="Green">E34</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 <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">r</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><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 NAME="E1:13_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><b>hence </b><a NAME="E2:13_1_1"/>
0 <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="E5:13_1"/>
( 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 <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></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>thus </b><a NAME="E2:13"/>
(  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">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 <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> ) ) implies <font color="Maroon">s</font> is <a href="comseq_2.html#V1" target="_self">bounded</a> )
 <b>by </b><i/>;<br/>


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