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

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="csspace.html#NM1" title="CSSPACE:NM.1">ComplexUnitarySpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">seq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">seq</font> is <a href="clvect_2.html#V1" title="CLVECT_2:attr.1">convergent</a> &amp;  <a href="clvect_2.html#K1" title="CLVECT_2:func.1">lim</a> <font color="Olive" title="b2">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> holds <br/>( <span class="p1"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Olive" title="b2">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</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_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Olive" title="b2">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><font color="Olive" title="b1">g</font></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> )</span><br/><b>let </b><font color="Maroon" title="c2">g</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">seq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">seq</font> is <a href="clvect_2.html#V1" title="CLVECT_2:attr.1">convergent</a> &amp;  <a href="clvect_2.html#K1" title="CLVECT_2:func.1">lim</a> <font color="Olive" title="b1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> holds <br/>( <span class="p1"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Olive" title="b1">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</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_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Olive" title="b1">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><font color="Maroon" title="c2">g</font></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> )</span><br/><b>let </b><font color="Maroon" title="c3">seq</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">seq</font> is <a href="clvect_2.html#V1" title="CLVECT_2:attr.1">convergent</a> &amp;  <a href="clvect_2.html#K1" title="CLVECT_2:func.1">lim</a> <font color="Maroon" title="c3">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> implies ( <span class="p1"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</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_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><font color="Maroon" title="c2">g</font></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> ) )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:24"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c3">seq</font> is <a href="clvect_2.html#V1" title="CLVECT_2:attr.1">convergent</a>
 <b>and </b><br/><a NAME="E2:24"/><i><font color="Green" title="E23">A2</font></i>: 
 <a href="clvect_2.html#K1" title="CLVECT_2:func.1">lim</a> <font color="Maroon" title="c3">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</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_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><font color="Maroon" title="c2">g</font></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> )</span><br/>

<a NAME="E3:24"/><i><font color="Green" title="E24">A3</font></i>: 
<span class="p1"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="clvect_2.html#T20" target="_self" title="CLVECT_2:th.20">Th20</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  ex <font color="Olive" title="b1">k</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 <br/>for <font color="Olive" title="b2">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">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> holds <br/> <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_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><font color="Maroon" title="c2">g</font></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> )</span><br/><a NAME="E1:24_1"/><i><font color="Green" title="E25">A4</font></i>: 
<font color="Maroon" title="c4">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="E2:24_1"/>
<font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">k</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 <br/>for <font color="Olive" title="b2">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">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> holds <br/> <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_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><font color="Maroon" title="c2">g</font></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font></span><br/><b>then consider </b><font color="Maroon" title="c5">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="E4:24_1"/><i><font color="Green" title="E26">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="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">m1</font> holds <br/><span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">seq</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#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">g</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="#E2:24"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="#E1:24_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="clvect_2.html#T19" target="_self" title="CLVECT_2:th.19">Th19</a></i>;<br/><b>take </b><font color="Maroon" title="c6">k</font> = <font color="Maroon" title="c5">m1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/> <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_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><font color="Maroon" title="c2">g</font></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">k</font> implies  <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_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><font color="Maroon" title="c2">g</font></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> )</span><br/><b>assume </b><a NAME="E1:24_1_1"/>
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><font color="Maroon" title="c2">g</font></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font></span><br/><a NAME="E2:24_1_1"/><b>then </b><i><font color="Green" title="E27">A6</font></i>: 
<span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">g</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="txt" href="#E4:24_1"><i><font color="Green" title="E26">A5</font></i></a></i>;<br/><a NAME="E3:24_1_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><font color="Maroon" title="c2">g</font></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">g</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span>
 <b>by </b><i><a class="ref" href="csspace.html#T51" title="CSSPACE:th.51">CSSPACE:51</a></i>;<br/><a NAME="E4:24_1_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="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><font color="Maroon" title="c2">g</font></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="txt" href="#E2:24_1_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:24_1_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_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><font color="Maroon" title="c2">g</font></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="ref" href="clvect_2.html#D3" target="_self" title="CLVECT_2:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E6:24_1"/>
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="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/> <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_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><font color="Maroon" title="c2">g</font></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:24"/>
( <span class="p1"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</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_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><font color="Maroon" title="c2">g</font></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> )
 <b>by </b><i><a class="txt" href="#E3:24"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
