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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="E1:29"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">seq</font> is <a href="seqm_3.html#V3">non-decreasing</a> &amp; <font color="Maroon">seq</font> is <a href="seq_2.html#V1">bounded_above</a> )
 ;<br/>

<a NAME="E2:29"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a> )
 
<b>by </b><i>, <a class="ref" href="seq_4.html#T51">SEQ_4:51</a></i>;<br/>
<a NAME="E3:29"/><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="E4:29"/><b>then </b>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K1" target="_self">sup</a> <font color="Maroon">seq</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:29"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K1" target="_self">sup</a> <font color="Maroon">seq</font>
 
<b>by </b><i>, <a class="ref" href="prepower.html#T3">PREPOWER:3</a></i>;<br/>
<a NAME="E6:29"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font>
 
<b>by </b><i>, <a class="ref" href="seq_4.html#T54">SEQ_4:54</a></i>;<br/>
<a NAME="E7:29"/><b>then </b>
 <a href="rinfsup1.html#K1" target="_self">sup</a> <font color="Maroon">seq</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:29"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a>  <a href="rinfsup1.html#K1" target="_self">sup</a> <font color="Maroon">seq</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


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