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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:50"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:50"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="bhsp_3.html#M1">subsequence</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="bhsp_3.html#M1">subsequence</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="frechet2.html#T3" target="_self">Th3</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>thus </b><a NAME="E2:50_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="frechet2.html#E1:50"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:50"/>
<font color="Maroon">c<sub>3</sub></font> <a href="frechet2.html#R1" target="_self">is_a_cluster_point_of</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="frechet2.html#T34" target="_self">Th34</a></i>;<br/>


</div>
