<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>(  <a href="irrat_1.html#K2" target="_self">bseq</a> <font color="Maroon">a<sub>1</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K2" target="_self">bseq</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="newton.html#K12">!</a> </span>)</span> );<br/>
<a NAME="E1:19"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">bseq0</font> = <a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> 1 as   <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E2:19_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">bseq0</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E3:19_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">bseq0</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">bseq0</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:19_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/><span class="p1">(<span class="default"><a href="irrat_1.html#K2" target="_self">bseq</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bseq0</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1"><b>proof </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>assume </b><a NAME="E1:19_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="irrat_1.html#K2" target="_self">bseq</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
1
<b>by </b><i/>
<br/>.= 
<font color="Maroon">bseq0</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:19_1"/>
 <a href="irrat_1.html#K2" target="_self">bseq</a> 0 is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i>, <a class="ref" href="seq_4.html#T31">SEQ_4:31</a></i>;<br/>

<b>thus </b><a NAME="E6:19_1"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K2" target="_self">bseq</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">0 <a href="newton.html#K12">!</a> </span>)</span>
 <b>by </b><i>, , <a class="ref" href="newton.html#T18">NEWTON:18</a>, <a class="ref" href="seq_4.html#T32">SEQ_4:32</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><b>thus </b><a NAME="E2:19_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="irrat_1.html#K2" target="_self">bseq</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">seq</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E2:19_2_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<a NAME="E3:19_2_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K2" target="_self">bseq</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>;<br/>
<a NAME="E4:19_2_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:19_2_1_1"/>
<font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i>, , <a class="ref" href="seq_2.html#T21">SEQ_2:21</a></i>;<br/>

<b>thus </b> <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font> = 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="seq_2.html#T22">SEQ_2:22</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="irrat_1.html#K1" target="_self">aseq</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">seq1</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E6:19_2_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<a NAME="E7:19_2_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">seq1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq</font> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K1" target="_self">aseq</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>;<br/>
<a NAME="E8:19_2_1"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">seq1</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq1</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:19_2_1_2"/><i><font color="Green">E78</font></i>: 
(  <a href="irrat_1.html#K1" target="_self">aseq</a> <font color="Maroon">k</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K1" target="_self">aseq</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:19_2_1_2"/>
<font color="Maroon">seq1</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i>, , <a class="ref" href="seq_2.html#T28">SEQ_2:28</a></i>;<br/>

<b>thus </b> <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq1</font> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="irrat_1.html#K1" target="_self">aseq</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="ref" href="seq_2.html#T29">SEQ_2:29</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span>
<b>by </b><i>, </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E9:19_2_1"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/><span class="p1">(<span class="default"><a href="irrat_1.html#K2" target="_self">bseq</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_3"><b>proof </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>assume </b><a NAME="E1:19_2_1_3"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<b>hence </b><span class="p1">(<span class="default"><a href="irrat_1.html#K2" target="_self">bseq</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="irrat_1.html#K2" target="_self">bseq</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="irrat_1.html#K1" target="_self">aseq</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="irrat_1.html#K1" target="_self">aseq</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i/>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:19_2_1"/>
 <a href="irrat_1.html#K2" target="_self">bseq</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i>, <a class="ref" href="seq_4.html#T31">SEQ_4:31</a></i>;<br/>

<b>thus </b><a NAME="E11:19_2_1"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K2" target="_self">bseq</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span>
 <b>by </b><i>, , <a class="ref" href="seq_4.html#T32">SEQ_4:32</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:19"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E4:19"/>
(  <a href="irrat_1.html#K2" target="_self">bseq</a> <font color="Maroon">k</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K2" target="_self">bseq</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="newton.html#K12">!</a> </span>)</span> )
 ;<br/>

<b>hence </b><a NAME="E5:19"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K2" target="_self">bseq</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="irrat_1.html#K5" target="_self">eseq</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/>


</div>
