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<b>let </b><font color="Maroon">X</font> be   <a href="csspace.html#NM1">ComplexUnitarySpace</a>;<br/><b>let </b><font color="Maroon">seq1</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">seq2</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:14"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">seq1</font> is <a href="clvect_2.html#V1" target="_self">convergent</a>
 <b>and </b><br/><a NAME="E2:14"/><i><font color="Green">E44</font></i>: 
 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">n</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="Olive">k</font> holds <br/><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font>
 ;<br/>

<a NAME="E3:14"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">seq2</font> is <a href="clvect_2.html#V1" target="_self">convergent</a>
 
<b>by </b><i>, , </i>;<br/>
<b>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="E5:14"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">n</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> holds <br/><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font>
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>now </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="E1:14_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>then consider </b><font color="Maroon">m1</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="E3:14_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">n</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">m1</font> holds <br/> <a href="csspace.html#K15">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">seq1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/><div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font>
 ;<br/><b>take </b><font color="Maroon">m</font> = <font color="Maroon">m1</font>;<br/><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>assume </b><a NAME="E2:14_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E3:14_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:14_1_1"/><b>then </b>
<font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:14_1_1"/>
 <a href="csspace.html#K15">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">seq1</font></span>)</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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><b>take </b><font color="Maroon">m</font> = <font color="Maroon">k</font>;<br/><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>assume </b><a NAME="E2:14_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E3:14_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m1</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:14_1_2"/><b>then </b>
 <a href="csspace.html#K15">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">seq1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:14_1_2"/>
 <a href="csspace.html#K15">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">seq1</font></span>)</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="E6:14_1"/>
 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">n</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="Olive">m</font> holds <br/> <a href="csspace.html#K15">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">seq1</font></span>)</span> <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="E7:14"/>
 <a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">seq1</font> <a href="hidden.html#R1">=</a>  <a href="clvect_2.html#K1" target="_self">lim</a> <font color="Maroon">seq2</font>
 <b>by </b><i>, </i>;<br/>


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