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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</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="nat_1.html#NM1">Nat</a>;<br/>





<b>assume </b><a NAME="E1:52"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="bhsp_2.html#V1">convergent</a>
 ;<br/>

<a NAME="E2:52"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="bhsp_3.html#K2" target="_self">^\</a> <font color="Maroon">c<sub>3</sub></font> is    <a href="bhsp_3.html#M1" target="_self">subsequence</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="bhsp_3.html#T43" target="_self">Th43</a></i>;<br/>
<b>hence </b><a NAME="E3:52"/>
<font color="Maroon">c<sub>2</sub></font> <a href="bhsp_3.html#K2" target="_self">^\</a> <font color="Maroon">c<sub>3</sub></font> is <a href="bhsp_2.html#V1">convergent</a>
 <b>by </b><i><a class="txt" href="bhsp_3.html#E1:52"><i><font color="Green">E23</font></i></a>, <a class="ref" href="bhsp_3.html#T32" target="_self">Th32</a></i>;<br/>

<b>thus </b><a NAME="E4:52"/>
 <a href="bhsp_2.html#K1">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bhsp_3.html#K2" target="_self">^\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="bhsp_2.html#K1">lim</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="bhsp_3.html#E1:52"><i><font color="Green">E23</font></i></a>, <a class="txt" href="bhsp_3.html#E2:52"><i><font color="Green">E24</font></i></a>, <a class="ref" href="bhsp_3.html#T33" target="_self">Th33</a></i>;<br/>


</div>
