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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:66"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V3">bounded</a>
 ;<br/>

<a NAME="E2:66"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V1">bounded_above</a> &amp; <font color="Maroon">c<sub>1</sub></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="E3:66"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seqm_3.html#V3">non-decreasing</a>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T52" target="_self">Th52</a></i>;<br/>
<a NAME="E4:66"/>
 <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:66"><i><font color="Green">E61</font></i></a>, <a class="ref" href="rinfsup1.html#T58" target="_self">Th58</a></i>;<br/>
<a NAME="E5:66"/><b>then </b>
 <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V1">bounded_above</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="E6:66"/>
(  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>1</sub></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#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <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">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="rinfsup1.html#E3:66"><i><font color="Green">E62</font></i></a>, <a class="ref" href="rinfsup1.html#T24" target="_self">Th24</a>, <a class="ref" href="seq_4.html#T51">SEQ_4:51</a></i>;<br/>


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