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

<b>assume </b><a NAME="E1:67"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a>
 ;<br/>

<a NAME="E2:67"/><b>then </b>
( <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/>
<a NAME="E3:67"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font> is <a href="seqm_3.html#V4">non-increasing</a>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E4:67"/>
 <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T15" target="_self">Th15</a>, <a class="ref" href="rinfsup1.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E5:67"/><b>then </b>
 <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <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>hence </b><a NAME="E6:67"/>
(  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rinfsup1.html#K2" target="_self">inf</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span> )
 <b>by </b><i><a class="ref" href="rinfsup1.html#T16" target="_self">Th16</a>, , <a class="ref" href="seq_4.html#T52">SEQ_4:52</a></i>;<br/>


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