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<div><i><font color="Green" title="E27">A1</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" title="c1">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="irrat_1.html#K4" title="IRRAT_1:func.4">dseq</a>  <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="irrat_1.html#NK1" title="IRRAT_1:NK.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><b>thus </b><span class="p1">(<span class="default"><a href="irrat_1.html#K4" title="IRRAT_1:func.4">dseq</a>  <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font> = 
<a href="irrat_1.html#K4" title="IRRAT_1:func.4">dseq</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="irrat_1.html#NK1" title="IRRAT_1:NK.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="irrat_1.html#D4" target="_self" title="IRRAT_1:def.4">Def4</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E2:29"/><b>then </b><i><font color="Green" title="E28">A2</font></i>: 
<a href="irrat_1.html#K4" title="IRRAT_1:func.4">dseq</a>  <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1 is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="power.html#T67" title="POWER:th.67">POWER:67</a></i>;<br/>
<b>hence </b><a NAME="E3:29"/>
 <a href="irrat_1.html#K4" title="IRRAT_1:func.4">dseq</a>  is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_4.html#T35" title="SEQ_4:th.35">SEQ_4:35</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <a href="irrat_1.html#K4" title="IRRAT_1:func.4">dseq</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="power.html#K8" title="POWER:func.8">number_e</a> </span><br/>

<a NAME="E4:29"/>
 <a href="power.html#K8" title="POWER:func.8">number_e</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K4" title="IRRAT_1:func.4">dseq</a>  <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:29"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="power.html#D4" title="POWER:def.4">POWER:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:29"/>
 <a href="power.html#K8" title="POWER:func.8">number_e</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <a href="irrat_1.html#K4" title="IRRAT_1:func.4">dseq</a> 
 <b>by </b><i><a class="txt" href="#E2:29"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="seq_4.html#T36" title="SEQ_4:th.36">SEQ_4:36</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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