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<b>let </b><font color="Maroon" title="c1">a</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>consider </b><font color="Maroon" title="c2">s</font> being   <a href="prepower.html#NM1" title="PREPOWER:NM.1">Rational_Sequence</a><b> such that </b><br/><a NAME="E2:97"/><i><font color="Green" title="E77">A1</font></i>: 
( <font color="Maroon" title="c2">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s</font> &amp; ( 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="c2">s</font> <a href="prepower.html#K10" title="PREPOWER:func.10">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font> ) )
 <b>by </b><i><a class="ref" href="prepower.html#T79" target="_self" title="PREPOWER:th.79">Th79</a></i>;<br/>
<a NAME="E3:97"/><i><font color="Green" title="E78">A2</font></i>: 
1 <a href="prepower.html#K11" title="PREPOWER:func.11">#Q</a> <font color="Maroon" title="c2">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="prepower.html#E2:97"><i><font color="Green" title="E77">A1</font></i></a>, <a class="ref" href="prepower.html#T82" target="_self" title="PREPOWER:th.82">Th82</a></i>;<br/>
<div><i><font color="Green" title="E79">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">n</font> be    <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> ;<br/><b>thus </b><span class="p1">(<span class="default">1 <a href="prepower.html#K11" title="PREPOWER:func.11">#Q</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font> = 
1 <a href="prepower.html#K9" title="PREPOWER:func.9">#Q</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="prepower.html#K10" title="PREPOWER:func.10">.</a> <font color="Maroon" title="c3">n</font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D7" target="_self" title="PREPOWER:def.7">Def7</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="prepower.html#T62" target="_self" title="PREPOWER:th.62">Th62</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E5:97"/><b>then </b>
1 <a href="prepower.html#K11" title="PREPOWER:func.11">#Q</a> <font color="Maroon" title="c2">s</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D6" title="SEQM_3:def.6">SEQM_3:def 6</a></i>;<br/>
<b>then </b> <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default">1 <a href="prepower.html#K11" title="PREPOWER:func.11">#Q</a> <font color="Maroon" title="c2">s</font></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="prepower.html#K11" title="PREPOWER:func.11">#Q</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0
<b>by </b><i><a class="ref" href="seq_4.html#T41" title="SEQ_4:th.41">SEQ_4:41</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="prepower.html#E4:97"><i><font color="Green" title="E79">A3</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E7:97"/>
1 <a href="prepower.html#K12" title="PREPOWER:func.12">#R</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="prepower.html#E2:97"><i><font color="Green" title="E77">A1</font></i></a>, <a class="txt" href="prepower.html#E3:97"><i><font color="Green" title="E78">A2</font></i></a>, <a class="ref" href="prepower.html#D8" target="_self" title="PREPOWER:def.8">Def8</a></i>;<br/>


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