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<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="irrat_1.html#K4" target="_self">dseq</a>  <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<a href="irrat_1.html#K4" target="_self">dseq</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="irrat_1.html#NK1" target="_self">^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<a NAME="E2:29"/><b>then </b><i><font color="Green">E41</font></i>: 
<a href="irrat_1.html#K4" target="_self">dseq</a>  <a href="seqm_3.html#K1">^\</a> 1 is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="power.html#T67">POWER:67</a></i>;<br/>
<b>hence </b><a NAME="E3:29"/>
 <a href="irrat_1.html#K4" target="_self">dseq</a>  is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_4.html#T35">SEQ_4:35</a></i>;<br/>

<a NAME="E4:29"/>
 <a href="power.html#K8">number_e</a>  <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K4" target="_self">dseq</a>  <a href="seqm_3.html#K1">^\</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="power.html#D4">POWER:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:29"/>
 <a href="power.html#K8">number_e</a>  <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <a href="irrat_1.html#K4" target="_self">dseq</a> 
 <b>by </b><i>, <a class="ref" href="seq_4.html#T36">SEQ_4:36</a></i>;<br/>


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