<?xml version="1.0"?>
<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>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:27"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a>
 ;<br/>

<a NAME="E2:27"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">seq</font>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T47">SEQM_3:47</a></i>;<br/>
<b>hence </b><a NAME="E3:27"/>
<font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E4:27"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font>
 <b>by </b><i>, , </i>;<br/>


</div>
