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<b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:22"/><i><font color="Green">E45</font></i>: 
 <a href="seq_1.html#K22">abs</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a>
 <b>and </b><br/><a NAME="E2:22"/><i><font color="Green">E46</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E3:22"/><i><font color="Green">E48</font></i>: 
 <a href="seq_1.html#K17">-</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq</font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, <a class="ref" href="seq_2.html#T23">SEQ_2:23</a></i>;<br/>
<a NAME="E4:22"/><i><font color="Green">E49</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="seq_2.html#T24">SEQ_2:24</a></i>;<br/>
<div><i><font color="Green">E50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>now </b></a><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/><a NAME="E1:22_1"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/><a NAME="E2:22_1"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>;<br/><a NAME="E3:22_1"/>
<font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/><b>hence </b><a NAME="E4:22_1"/>
( <span class="p1">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="seq_1.html#T14">SEQ_1:14</a>, <a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:22"/>
<font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i>, , , , <a class="ref" href="seq_2.html#T33">SEQ_2:33</a></i>;<br/>

<b>thus </b><a NAME="E7:22"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , , , , <a class="ref" href="seq_2.html#T34">SEQ_2:34</a></i>;<br/>


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