<?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">A1</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>assume </b><a NAME="E1:73"/>
<font color="Maroon" title="c2">A1</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-decreasing</a>
 ;<br/>

<a NAME="E2:73"/><b>then </b><i><font color="Green" title="E54">A1</font></i>: 
(  <a href="kurato_2.html#K5" title="KURATO_2:func.5">lim_sup</a> <font color="Maroon" title="c2">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c2">A1</font> &amp;  <a href="kurato_2.html#K4" title="KURATO_2:func.4">lim_inf</a> <font color="Maroon" title="c2">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c2">A1</font> )
 
<b>by </b><i><a class="ref" href="setlim_1.html#T49" target="_self" title="SETLIM_1:th.49">Th49</a>, <a class="ref" href="setlim_1.html#T52" target="_self" title="SETLIM_1:th.52">Th52</a></i>;<br/>
<b>hence </b><a NAME="E3:73"/>
<font color="Maroon" title="c2">A1</font> is <a href="kurato_2.html#V3" title="KURATO_2:attr.3">convergent</a>
 <b>by </b><i><a class="ref" href="kurato_2.html#D7" title="KURATO_2:def.7">KURATO_2:def 7</a></i>;<br/>

<b>hence </b><a NAME="E4:73"/>
 <a href="setlim_1.html#K3" title="SETLIM_1:func.3">lim</a> <font color="Maroon" title="c2">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c2">A1</font>
 <b>by </b><i><a class="txt" href="setlim_1.html#E2:73"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="setlim_1.html#D6" target="_self" title="SETLIM_1:def.6">Def6</a></i>;<br/>


</div>
