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

<b>let </b><font color="Maroon" title="c1">s</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon" title="c2">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>assume </b><a NAME="E1:39"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ;<br/>

<b>assume </b><a NAME="E2:39"/><i><font color="Green" title="E33">A2</font></i>: 
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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/><font color="Maroon" title="c1">s</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <font color="Maroon" title="c2">a</font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <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> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c3">s1</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><b> such that </b><br/><a NAME="E4:39"/><i><font color="Green" title="E34">A3</font></i>: 
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="c3">s1</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1" title="SEQ_1:sch.1">SEQ_1:sch 1</a>();<br/></i>
<a NAME="E5:39"/><i><font color="Green" title="E35">A4</font></i>: 
( <font color="Maroon" title="c3">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="prepower.html#E4:39"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="seq_4.html#T46" title="SEQ_4:th.46">SEQ_4:46</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">s2</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> 1 as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> <b>by </b><i><a class="ref" href="funcop_1.html#T57" title="FUNCOP_1:th.57">FUNCOP_1:57</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">s3</font> = <font color="Maroon" title="c4">s2</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c3">s1</font>;<br/>
<a NAME="E7:39"/><i><font color="Green" title="E36">A5</font></i>: 
<font color="Maroon" title="c4">s2</font> 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#T39" title="SEQ_4:th.39">SEQ_4:39</a></i>;<br/>
<a NAME="E8:39"/><b>then </b><i><font color="Green" title="E37">A6</font></i>: 
<font color="Maroon" title="c4">s2</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c3">s1</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#E5:39"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="seq_2.html#T19" title="SEQ_2:th.19">SEQ_2:19</a></i>;<br/>
<i><font color="Green" title="E38">A7</font></i>:  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s2</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c3">s1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">s2</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="prepower.html#E5:39"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="seq_4.html#T59" title="SEQ_4:th.59">SEQ_4:59</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/>
<i><font color="Green" title="E39">A8</font></i>:  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s2</font> = 
<font color="Maroon" title="c4">s2</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<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="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/>
<div><i><font color="Green" title="E40">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">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/><a NAME="E1:39_3"/><i><font color="Green" title="E41">A10</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><i><font color="Green" title="E42">A11</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font> = 
<font color="Maroon" title="c1">s</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="prepower.html#E2:39"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="prepower.html#E1:39_3"><i><font color="Green" title="E41">A10</font></i></a></i>
;<br/><a NAME="E3:39_3"/><b>then </b><i><font color="Green" title="E43">A12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="prepower.html#E1:39"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="prepower.html#E1:39_3"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="prepower.html#T38" target="_self" title="PREPOWER:th.38">Th38</a></i>;<br/><a NAME="E4:39_3"/><b>then </b><i><font color="Green" title="E44">A13</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/><b>set </b><font color="Maroon" title="c7">b</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1;<br/><a NAME="E5:39_3"/><i><font color="Green" title="E45">A14</font></i>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="prepower.html#E4:39_3"><i><font color="Green" title="E44">A13</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:39_3"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="prepower.html#E1:39"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="prepower.html#E1:39_3"><i><font color="Green" title="E41">A10</font></i></a>, <a class="txt" href="prepower.html#E2:39_3"><i><font color="Green" title="E42">A11</font></i></a>, <a class="txt" href="prepower.html#E31"><i><font color="Green" title="E18">Lm2</font></i></a></i>;<br/><a NAME="E7:39_3"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="prepower.html#E5:39_3"><i><font color="Green" title="E45">A14</font></i></a>, <a class="ref" href="prepower.html#T24" target="_self" title="PREPOWER:th.24">Th24</a></i>;<br/><a NAME="E8:39_3"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E9:39_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E10:39_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 ;<br/><a NAME="E11:39_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E12:39_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>;<br/><a NAME="E13:39_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E14:39_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">s2</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><a NAME="E15:39_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">s2</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="prepower.html#E4:39"><i><font color="Green" title="E34">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E16:39_3"/>
( <font color="Maroon" title="c4">s2</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s2</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font> )
 <b>by </b><i><a class="txt" href="prepower.html#E3:39_3"><i><font color="Green" title="E43">A12</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a>, <a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:39"/><b>then </b><i><font color="Green" title="E41">A15</font></i>: 
<font color="Maroon" title="c1">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1 is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="prepower.html#E7:39"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="prepower.html#E8:39"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="prepower.html#E9:39"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="prepower.html#E10:39"><i><font color="Green" title="E39">A8</font></i></a>, <a class="ref" href="seq_2.html#T33" title="SEQ_2:th.33">SEQ_2:33</a></i>;<br/>
<a NAME="E13:39"/><i><font color="Green" title="E42">A16</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="prepower.html#E7:39"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="prepower.html#E8:39"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="prepower.html#E9:39"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="prepower.html#E10:39"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="prepower.html#E11:39"><i><font color="Green" title="E40">A9</font></i></a>, <a class="ref" href="seq_2.html#T34" title="SEQ_2:th.34">SEQ_2:34</a></i>;<br/>
<b>thus </b><a NAME="E14:39"/>
<font color="Maroon" title="c1">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#E12:39"><i><font color="Green" title="E41">A15</font></i></a>, <a class="ref" href="seq_4.html#T35" title="SEQ_4:th.35">SEQ_4:35</a></i>;<br/>

<b>thus </b><a NAME="E15:39"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="prepower.html#E12:39"><i><font color="Green" title="E41">A15</font></i></a>, <a class="txt" href="prepower.html#E13:39"><i><font color="Green" title="E42">A16</font></i></a>, <a class="ref" href="seq_4.html#T36" title="SEQ_4:th.36">SEQ_4:36</a></i>;<br/>


</div>
