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<b>let </b><font color="Maroon" title="c1">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:29"/><i><font color="Green" title="E26">A1</font></i>: 
( <font color="Maroon" title="c1">seq</font> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a> &amp; <font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> )
 ;<br/>

<a NAME="E2:29"/><b>then </b><i><font color="Green" title="E27">A2</font></i>: 
( <font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp; <font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T22" target="_self" title="RINFSUP1:th.22">Th22</a>, <a class="ref" href="seq_4.html#T51" title="SEQ_4:th.51">SEQ_4:51</a></i>;<br/>
<a NAME="E3:29"/><b>then </b>
( <font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> &amp; <font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="seq_2.html#D5" title="SEQ_2:def.5">SEQ_2:def 5</a></i>;<br/>
<a NAME="E4:29"/><b>then </b>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c1">seq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c1">seq</font>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T7" target="_self" title="RINFSUP1:th.7">Th7</a></i>;<br/>
<a NAME="E5:29"/><b>then </b><i><font color="Green" title="E28">A3</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c1">seq</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c1">seq</font>
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E2:29"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="prepower.html#T3" title="PREPOWER:th.3">PREPOWER:3</a></i>;<br/>
<a NAME="E6:29"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c1">seq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c1">seq</font>
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:29"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="seq_4.html#T54" title="SEQ_4:th.54">SEQ_4:54</a></i>;<br/>
<a NAME="E7:29"/><b>then </b>
 <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c1">seq</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c1">seq</font>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T9" target="_self" title="RINFSUP1:th.9">Th9</a></i>;<br/>
<b>hence </b><a NAME="E8:29"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c1">seq</font>
 <b>by </b><i><a class="txt" href="rinfsup1.html#E5:29"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


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