<?xml version="1.0"?>
<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>let </b><font color="Maroon">vseq</font> be   <a href="normsp_1.html#NM2">sequence</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:22"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">vseq</font> is <a href="rsspace3.html#V1">CCauchy</a>
 ;<br/>

<a NAME="E2:22"/><i><font color="Green">E34</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">xvseq</font> = <font color="Maroon">vseq</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">rseqi</font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>for <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Olive">rseqi</font> <a href="seq_1.html#K2">.</a> <font color="Olive">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xvseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Olive">l</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">rseqi</font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Olive">rseqi</font> );<br/>
<a NAME="E4:22"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/> ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:22_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/>

<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="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">xvseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>;<br/>
<b>consider </b><font color="Maroon">rseqi</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E3:22_1"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">rseqi</font> <a href="seq_1.html#K2">.</a> <font color="Olive">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<a NAME="E4:22_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">rseqi</font> is <a href="seq_2.html#V4">convergent</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:22_1_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>thus </b><a NAME="E2:22_1_1_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">rseqi</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">rseqi</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:22_1_1_1_1"/>
<font color="Maroon">e</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:22_1_1_1_1"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">m</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> <font color="Maroon">k</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, , <a class="ref" href="rsspace3.html#T10">RSSPACE3:10</a></i>;<br/>
<b>take </b>
<font color="Maroon">k</font>
;<br/>

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

<b>assume </b><a NAME="E4:22_1_1_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E5:22_1_1_1_1"/><i><font color="Green">E73</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:22_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xvseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K8">-</a> <span class="p2">(<span class="default"><font color="Maroon">xvseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T2">EUCLID:2</a></i>;<br/>
<a NAME="E7:22_1_1_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xvseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K8">-</a> <span class="p2">(<span class="default"><font color="Maroon">xvseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, </i>;<br/>
<a NAME="E8:22_1_1_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xvseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xvseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">xvseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K8">-</a> <span class="p2">(<span class="default"><font color="Maroon">xvseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T47">RVSUM_1:47</a></i>;<br/>
<a NAME="E9:22_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">xvseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K8">-</a> <span class="p1">(<span class="default"><font color="Maroon">xvseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E10:22_1_1_1_1"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xvseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">xvseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">vseq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E11:22_1_1_1_1"/>
( <font color="Maroon">rseqi</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xvseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">rseqi</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xvseq</font> <a href="real_ns1.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E12:22_1_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">rseqi</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">rseqi</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="txt" href="real_ns1.html#E1"><i><font color="Green">Lemma28</font></i></a>, </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:22_1_1"/>
<font color="Maroon">rseqi</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_4.html#T58">SEQ_4:58</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">rseqi</font>
;<br/>

<b>thus </b><a NAME="E5:22_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>, <a href="seq_2.html#K2">lim</a> <font color="Maroon">rseqi</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E6:22"/><i><font color="Green">E78</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>();<br/></i>
<b>reconsider </b><font color="Maroon">tseq</font> = <font color="Maroon">f</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">xseq</font> = <font color="Maroon">tseq</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E9:22"/>
( <font color="Maroon">xseq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">tseq</font> &amp; <font color="Maroon">xvseq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vseq</font> )
 
;<br/>
<b>hence </b><a NAME="E10:22"/>
<font color="Maroon">vseq</font> is <a href="normsp_1.html#V4">convergent</a>
 <b>by </b><i>, </i>;<br/>


</div>
