<?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">seq1</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>let </b><font color="Maroon">seq2</font> be   <a href="normsp_1.html#NM2">sequence</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:32"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">seq1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq2</font>
 ;<br/>

<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="euclid.html#K1">REAL</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">seq1</font> is <a href="normsp_1.html#V4">convergent</a>
 ;<br/><b>then consider </b><font color="Maroon">RNg</font> being   <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> such that </b><br/><a NAME="E3:32_2"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> holds <br/> ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>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">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">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">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">RNg</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font>
 <b>by </b><i><a class="ref" href="normsp_1.html#D9">NORMSP_1:def 9</a></i>;<br/><b>reconsider </b><font color="Maroon">RUg</font> = <font color="Maroon">RNg</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i/>;<br/><a NAME="E5:32_2"/>
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> holds <br/> ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>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">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> holds <br/> <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span>,<font color="Maroon">RUg</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:32_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/>

<b>then consider </b><font color="Maroon">m</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="E3:32_2_1"/><i><font color="Green">E66</font></i>: 
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="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">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">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">RNg</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="real_ns1.html#E1"><i><font color="Green">Lemma28</font></i></a></i>;<br/>
<a NAME="E4:32_2_1"/><i><font color="Green">E67</font></i>: 
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="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> holds <br/> <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span>,<font color="Maroon">RUg</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1_1"><b>proof </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="E1:32_2_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">p</font> = <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">RNg</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/>;<br/>
<a NAME="E3:32_2_1_1"/>
( <font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> &amp;  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">RNg</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">RUg</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:32_2_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">RUg</font>
 
<b>by </b><i/>;<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">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">RNg</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> = 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">p</font></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i/>
<br/>.= 
 <a href="square_1.html#K6">sqrt</a> <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K1">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_2.html#T13">EUCLID_2:13</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">p</font>,<font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_2.html#D1">EUCLID_2:def 1</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_ns1.html#K6" target="_self">Euclid_scalar</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">p</font>,<font color="Maroon">p</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">RUg</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E6:32_2_1_1"/>
 <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">RUg</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">m</font>
;<br/>

<b>thus </b><a NAME="E5:32_2_1"/>
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="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> holds <br/> <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span>,<font color="Maroon">RUg</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:32_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">seq2</font> is <a href="bhsp_2.html#V1">convergent</a>
 <b>by </b><i><a class="ref" href="bhsp_2.html#D1">BHSP_2:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">LIMIT</font> =  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">seq1</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i/>;<br/><a NAME="E8:32_2"/>
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>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="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">m</font> holds <br/> <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span>,<font color="Maroon">LIMIT</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:32_2_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon">m</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="E3:32_2_2"/><i><font color="Green">E73</font></i>: 
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="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">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">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">seq1</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">r</font>
 <b>by </b><i>, <a class="ref" href="normsp_1.html#D11">NORMSP_1:def 11</a></i>;<br/>
<a NAME="E4:32_2_2"/><i><font color="Green">E74</font></i>: 
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="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> holds <br/> <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span>,<font color="Maroon">LIMIT</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_2_1"><b>proof </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="E1:32_2_2_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">p</font> = <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">seq1</font></span>)</span> 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/>;<br/>
<a NAME="E3:32_2_2_1"/>
( <font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> &amp;  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">seq1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">LIMIT</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:32_2_2_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">LIMIT</font>
 
<b>by </b><i/>;<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">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">seq1</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> = 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">p</font></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i/>
<br/>.= 
 <a href="square_1.html#K6">sqrt</a> <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K1">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_2.html#T13">EUCLID_2:13</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">p</font>,<font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_2.html#D1">EUCLID_2:def 1</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_ns1.html#K6" target="_self">Euclid_scalar</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">p</font>,<font color="Maroon">p</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">LIMIT</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span>
<b>by </b><i><a class="txt" href="real_ns1.html#E1:32"><i><font color="Green">E33</font></i></a>, </i>
;<br/>
<b>hence </b><a NAME="E6:32_2_2_1"/>
 <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">LIMIT</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">m</font>
;<br/>

<b>thus </b><a NAME="E5:32_2_2"/>
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="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> holds <br/> <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span>,<font color="Maroon">LIMIT</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:32_2"/>
 <a href="bhsp_2.html#K1">lim</a> <font color="Maroon">seq2</font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">seq1</font>
 <b>by </b><i>, <a class="ref" href="bhsp_2.html#D2">BHSP_2:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:32"/>
( <font color="Maroon">seq1</font> is <a href="normsp_1.html#V4">convergent</a> implies ( <font color="Maroon">seq2</font> is <a href="bhsp_2.html#V1">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">seq1</font> <a href="hidden.html#R1">=</a>  <a href="bhsp_2.html#K1">lim</a> <font color="Maroon">seq2</font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_3"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">seq2</font> is <a href="bhsp_2.html#V1">convergent</a>
 ;<br/><b>then consider </b><font color="Maroon">RUg</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:32_3"/><i><font color="Green">E88</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> holds <br/> ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>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">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> holds <br/> <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span>,<font color="Maroon">RUg</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font>
 <b>by </b><i><a class="ref" href="bhsp_2.html#D1">BHSP_2:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">RNg</font> = <font color="Maroon">RUg</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="E5:32_3"/>
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> holds <br/> ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>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">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">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">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">RNg</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:32_3_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/>

<b>then consider </b><font color="Maroon">m</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="E3:32_3_1"/><i><font color="Green">E89</font></i>: 
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="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> holds <br/> <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span>,<font color="Maroon">RUg</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="real_ns1.html#E1"><i><font color="Green">Lemma28</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">m</font>
;<br/>

<a NAME="E4:32_3_1"/>
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="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">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">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">RNg</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1"><b>proof </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="E1:32_3_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E2:32_3_1_1"/><b>then </b><i><font color="Green">E90</font></i>: 
 <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">RUg</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">RUg</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/>;<br/>
<a NAME="E4:32_3_1_1"/>
( <font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> &amp;  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">RNg</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">RUg</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:32_3_1_1"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">RNg</font>
 
<b>by </b><i/>;<br/>
 <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">RUg</font> = 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_ns1.html#K6" target="_self">Euclid_scalar</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">p</font>,<font color="Maroon">p</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">p</font>,<font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="square_1.html#K6">sqrt</a> <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K1">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_2.html#D1">EUCLID_2:def 1</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">p</font></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid_2.html#T13">EUCLID_2:13</a></i>
;<br/>
<b>hence </b><a NAME="E7:32_3_1_1"/>
<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">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">RNg</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:32_3_1"/>
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="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">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">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">RNg</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:32_3"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">seq1</font> is <a href="normsp_1.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="normsp_1.html#D9">NORMSP_1:def 9</a></i>;<br/><b>reconsider </b><font color="Maroon">LIMIT</font> =  <a href="bhsp_2.html#K1">lim</a> <font color="Maroon">seq2</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="E8:32_3"/>
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>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="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">m</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">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">LIMIT</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:32_3_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon">m</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="E3:32_3_2"/><i><font color="Green">E106</font></i>: 
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="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> holds <br/> <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="bhsp_2.html#K1">lim</a> <font color="Maroon">seq2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="bhsp_2.html#D2">BHSP_2:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon">m</font>
;<br/>

<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="E4:32_3_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E5:32_3_2"/><b>then </b><i><font color="Green">E108</font></i>: 
 <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="bhsp_2.html#K1">lim</a> <font color="Maroon">seq2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="bhsp_2.html#K1">lim</a> <font color="Maroon">seq2</font></span>)</span> 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/>;<br/>
<a NAME="E7:32_3_2"/>
( <font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> &amp;  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="bhsp_2.html#K1">lim</a> <font color="Maroon">seq2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">LIMIT</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:32_3_2"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">LIMIT</font>
 
<b>by </b><i/>;<br/>
 <a href="bhsp_1.html#K5">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="bhsp_2.html#K1">lim</a> <font color="Maroon">seq2</font></span>)</span> = 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_ns1.html#K6" target="_self">Euclid_scalar</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">p</font>,<font color="Maroon">p</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">p</font>,<font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="square_1.html#K6">sqrt</a> <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K1">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_2.html#D1">EUCLID_2:def 1</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">p</font></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid_2.html#T13">EUCLID_2:13</a></i>
;<br/>
<b>hence </b><a NAME="E10:32_3_2"/>
<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">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">LIMIT</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:32_3"/>
 <a href="normsp_1.html#K8">lim</a> <font color="Maroon">seq1</font> <a href="hidden.html#R1">=</a>  <a href="bhsp_2.html#K1">lim</a> <font color="Maroon">seq2</font>
 <b>by </b><i>, <a class="ref" href="normsp_1.html#D11">NORMSP_1:def 11</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:32"/>
( <font color="Maroon">seq2</font> is <a href="bhsp_2.html#V1">convergent</a> implies ( <font color="Maroon">seq1</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">seq1</font> <a href="hidden.html#R1">=</a>  <a href="bhsp_2.html#K1">lim</a> <font color="Maroon">seq2</font> ) )
 ;<br/>


</div>
