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<b>let </b><font color="Maroon" title="c4">CNS</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c5">g</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">CNS</font>;<br/><b>let </b><font color="Maroon" title="c6">S</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c4">CNS</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:146"/><i><font color="Green" title="E91">A1</font></i>: 
<font color="Maroon" title="c6">S</font> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a>
 <b>and </b><br/><a NAME="E2:146"/><i><font color="Green" title="E92">A2</font></i>: 
 <a href="clvect_1.html#K8" title="CLVECT_1:func.8">lim</a> <font color="Maroon" title="c6">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font>
 ;<br/>

<a NAME="E3:146"/>
<font color="Maroon" title="c6">S</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c5">g</font> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a>
 
<b>by </b><i><a class="txt" href="clvect_1.html#E1:146"><i><font color="Green" title="E91">A1</font></i></a>, <a class="ref" href="clvect_1.html#T117" target="_self" title="CLVECT_1:th.117">Th117</a></i>;<br/>
<a NAME="E4:146"/><b>then </b><i><font color="Green" title="E93">A3</font></i>: 
<span class="p1"><a href="clvect_1.html#K7" title="CLVECT_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c5">g</font></span>)</span></span><a href="clvect_1.html#K7" title="CLVECT_1:func.7">.||</a></span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="clvect_1.html#T119" target="_self" title="CLVECT_1:th.119">Th119</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="146_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">r</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/><a NAME="E1:146_1"/><i><font color="Green" title="E94">A4</font></i>: 
<font color="Maroon" title="c7">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><b>assume </b><a NAME="E2:146_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">r</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c8">m1</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> <b> such that </b><br/><a NAME="E4:146_1"/><i><font color="Green" title="E95">A5</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="Maroon" title="c8">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c5">g</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="txt" href="clvect_1.html#E1:146"><i><font color="Green" title="E91">A1</font></i></a>, <a class="txt" href="clvect_1.html#E2:146"><i><font color="Green" title="E92">A2</font></i></a>, <a class="txt" href="clvect_1.html#E1:146_1"><i><font color="Green" title="E94">A4</font></i></a>, <a class="ref" href="clvect_1.html#D17" target="_self" title="CLVECT_1:def.17">Def17</a></i>;<br/><b>take </b><font color="Maroon" title="c9">k</font> = <font color="Maroon" title="c8">m1</font>;<br/><b>let </b><font color="Maroon" title="c10">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/><b>assume </b><a NAME="E5:146_1"/>
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n</font>
 ;<br/><a NAME="E6:146_1"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c5">g</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="txt" href="clvect_1.html#E4:146_1"><i><font color="Green" title="E95">A5</font></i></a></i>;<br/><a NAME="E7:146_1"/><b>then </b><i><font color="Green" title="E96">A6</font></i>: 
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">CNS</font>)</span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T26" title="RLVECT_1:th.26">RLVECT_1:26</a></i>;<br/><a NAME="E8:146_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c5">g</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">CNS</font>)</span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">CNS</font>)</span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
 <b>by </b><i><a class="ref" href="clvect_1.html#T111" target="_self" title="CLVECT_1:th.111">Th111</a></i>;<br/><a NAME="E9:146_1"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c5">g</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">CNS</font>)</span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="txt" href="clvect_1.html#E7:146_1"><i><font color="Green" title="E96">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:146_1"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c5">g</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="ref" href="clvect_1.html#D10" target="_self" title="CLVECT_1:def.10">Def10</a></i>;<br/><a NAME="E11:146_1"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="ref" href="normsp_1.html#D7" title="NORMSP_1:def.7">NORMSP_1:def 7</a></i>;<br/><b>hence </b><a NAME="E12:146_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K7" title="CLVECT_1:func.7">||.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c5">g</font></span>)</span></span><a href="clvect_1.html#K7" title="CLVECT_1:func.7">.||</a></span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="ref" href="clvect_1.html#D16" target="_self" title="CLVECT_1:def.16">Def16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:146"/>
( <span class="p1"><a href="clvect_1.html#K7" title="CLVECT_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c5">g</font></span>)</span></span><a href="clvect_1.html#K7" title="CLVECT_1:func.7">.||</a></span> 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> <span class="p1"><a href="clvect_1.html#K7" title="CLVECT_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c5">g</font></span>)</span></span><a href="clvect_1.html#K7" title="CLVECT_1:func.7">.||</a></span> <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="clvect_1.html#E4:146"><i><font color="Green" title="E93">A3</font></i></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></i>;<br/>


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