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

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">S</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>assume </b><a NAME="E2:24"/><i><font color="Green" title="E21">A2</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1"/>
( <font color="Maroon" title="c3">S</font> is <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a> or  not <font color="Maroon" title="c3">S</font> is <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1"><b>suppose </b></a><a NAME="E1:24_1_1"/>
<font color="Maroon" title="c3">S</font> is <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:24_1_1"/>
 <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="frechet2.html#E2:24"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="frechet.html#T26" title="FRECHET:th.26">FRECHET:26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2"><b>suppose </b></a><a NAME="E1:24_1_2"/>
not <font color="Maroon" title="c3">S</font> is <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a>
 ;<br/><div class="add"><a NAME="E2:24_1_2"/><b>then </b>
 <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="frechet2.html#T11" target="_self" title="FRECHET2:th.11">Th11</a></i>;<br/><b>hence </b><a NAME="E3:24_1_2"/>
 <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
