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

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">Si</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">S</font> be    <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of <font color="Maroon" title="c2">Si</font>;<br/>





<b>assume </b><a NAME="E1:103"/>
<font color="Maroon" title="c3">S</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-increasing</a>
 ;<br/>

<a NAME="E2:103"/><b>then </b><i><font color="Green" title="E70">A1</font></i>: 
(  <a href="setlim_1.html#K7" title="SETLIM_1:func.7">lim_sup</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c3">S</font> &amp;  <a href="setlim_1.html#K6" title="SETLIM_1:func.6">lim_inf</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c3">S</font> )
 
<b>by </b><i><a class="ref" href="setlim_1.html#T89" target="_self" title="SETLIM_1:th.89">Th89</a>, <a class="ref" href="setlim_1.html#T90" target="_self" title="SETLIM_1:th.90">Th90</a></i>;<br/>
<b>hence </b><a NAME="E3:103"/>
<font color="Maroon" title="c3">S</font> is <a href="setlim_1.html#V3" title="SETLIM_1:attr.3">convergent</a>
 <b>by </b><i><a class="ref" href="setlim_1.html#D12" target="_self" title="SETLIM_1:def.12">Def12</a></i>;<br/>

<b>hence </b><a NAME="E4:103"/>
 <a href="setlim_1.html#K8" title="SETLIM_1:func.8">lim</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c3">S</font>
 <b>by </b><i><a class="txt" href="setlim_1.html#E2:103"><i><font color="Green" title="E70">A1</font></i></a>, <a class="ref" href="setlim_1.html#D13" target="_self" title="SETLIM_1:def.13">Def13</a></i>;<br/>


</div>
