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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="E1:85"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">seq</font> is <a href="seqm_3.html#V4">non-increasing</a> &amp; <font color="Maroon">seq</font> is <a href="seq_2.html#V2">bounded_below</a> )
 ;<br/>

<a NAME="E2:85"/><b>then </b>
 <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font> is <a href="funct_1.html#V3">constant</a>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T25" target="_self">Th25</a></i>;<br/>
<b>then consider </b><font color="Maroon">r1</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="E4:85"/><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">r1</font></span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="seqm_3.html#T15">SEQM_3:15</a></i>;<br/>
<a NAME="E5:85"/>
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T27" target="_self">Th27</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:85"/><b>then </b><i><font color="Green">E40</font></i>: 
 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">r1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T9">RFUNCT_2:9</a></i>;<br/>
 <a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> = 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">seq</font></span>)</span></span><a href="seq_4.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="rinfsup1.html#T29" target="_self">Th29</a>, <a class="ref" href="rinfsup1.html#T27" target="_self">Th27</a>, <a class="ref" href="rinfsup1.html#T26" target="_self">Th26</a>, <a class="ref" href="rinfsup1.html#T25" target="_self">Th25</a></i>
<br/>.= 
 <a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">seq</font>
<b>by </b><i><a class="ref" href="seq_4.html#T22">SEQ_4:22</a></i>
;<br/>
<b>hence </b><a NAME="E8:85"/>
 <a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rinfsup1.html#K2" target="_self">inf</a> <font color="Maroon">seq</font>
 ;<br/>


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