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<span class="kw">let </span><font color="Maroon" title="c1">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">seq</font> is  <a href="valued_0.html#V9" title="VALUED_0:attr.9">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>  implies  <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">upper_bound</a> <font color="Maroon" title="c1">seq</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:29"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
( <font color="Maroon" title="c1">seq</font> is  <a href="valued_0.html#V9" title="VALUED_0:attr.9">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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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">upper_bound</a> <font color="Maroon" title="c1">seq</font></span><br/>

<a NAME="E2:29"/><span class="kw">then </span>
 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#K1" title="SEQ_1:func.1">.</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T37" title="SEQ_4:th.37">SEQ_4:37</a></span>;<br/>
<a NAME="E3:29"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A2</font></span>: 
 <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">upper_bound</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup1.html#T9" target="_self" title="RINFSUP1:th.9">Th9</a></span>;<br/>
<a NAME="E4:29"/>
 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#K1" title="SEQ_1:func.1">.</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">upper_bound</a> <font color="Maroon" title="c1">seq</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="rinfsup1.html#T7" target="_self" title="RINFSUP1:th.7">Th7</a></span>;<br/>
<a NAME="E5:29"/><span class="kw">then </span>
 <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">upper_bound</a> <font color="Maroon" title="c1">seq</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="prepower.html#T2" title="PREPOWER:th.2">PREPOWER:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:29"/>
 <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">upper_bound</a> <font color="Maroon" title="c1">seq</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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