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<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">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">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="E1:96"/><i><font color="Green" title="E80">A1</font></i>: 
<font color="Maroon" title="c3">A1</font> is <a href="kurato_2.html#V3" title="KURATO_2:attr.3">convergent</a>
 ;<br/>

<b>then </b><i><font color="Green" title="E81">A2</font></i>:  <a href="kurato_2.html#K4" title="KURATO_2:func.4">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="setlim_2.html#K7" title="SETLIM_2:func.7">(\)</a> <font color="Maroon" title="c3">A1</font></span>)</span> = 
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K4" title="KURATO_2:func.4">lim_inf</a> <font color="Maroon" title="c3">A1</font></span>)</span>
<b>by </b><i><a class="ref" href="setlim_2.html#T79" target="_self" title="SETLIM_2:th.79">Th79</a></i>
<br/>.= 
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K3" title="SETLIM_1:func.3">lim</a> <font color="Maroon" title="c3">A1</font></span>)</span>
<b>by </b><i><a class="txt" href="setlim_2.html#E1:96"><i><font color="Green" title="E80">A1</font></i></a>, <a class="ref" href="setlim_1.html#D6" title="SETLIM_1:def.6">SETLIM_1:def 6</a></i>
;<br/>
 <a href="kurato_2.html#K5" title="KURATO_2:func.5">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="setlim_2.html#K7" title="SETLIM_2:func.7">(\)</a> <font color="Maroon" title="c3">A1</font></span>)</span> = 
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K5" title="KURATO_2:func.5">lim_sup</a> <font color="Maroon" title="c3">A1</font></span>)</span>
<b>by </b><i><a class="txt" href="setlim_2.html#E1:96"><i><font color="Green" title="E80">A1</font></i></a>, <a class="ref" href="setlim_2.html#T86" target="_self" title="SETLIM_2:th.86">Th86</a></i>
<br/>.= 
 <a href="kurato_2.html#K4" title="KURATO_2:func.4">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="setlim_2.html#K7" title="SETLIM_2:func.7">(\)</a> <font color="Maroon" title="c3">A1</font></span>)</span>
<b>by </b><i><a class="txt" href="setlim_2.html#E1:96"><i><font color="Green" title="E80">A1</font></i></a>, <a class="txt" href="setlim_2.html#E2:96"><i><font color="Green" title="E81">A2</font></i></a>, <a class="ref" href="setlim_1.html#D6" title="SETLIM_1:def.6">SETLIM_1:def 6</a></i>
;<br/>
<a NAME="E4:96"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="setlim_2.html#K7" title="SETLIM_2:func.7">(\)</a> <font color="Maroon" title="c3">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="E5:96"/>
( <font color="Maroon" title="c2">A</font> <a href="setlim_2.html#K7" title="SETLIM_2:func.7">(\)</a> <font color="Maroon" title="c3">A1</font> is <a href="kurato_2.html#V3" title="KURATO_2:attr.3">convergent</a> &amp;  <a href="setlim_1.html#K3" title="SETLIM_1:func.3">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="setlim_2.html#K7" title="SETLIM_2:func.7">(\)</a> <font color="Maroon" title="c3">A1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K3" title="SETLIM_1:func.3">lim</a> <font color="Maroon" title="c3">A1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="setlim_2.html#E2:96"><i><font color="Green" title="E81">A2</font></i></a>, <a class="ref" href="setlim_1.html#D6" title="SETLIM_1:def.6">SETLIM_1:def 6</a></i>;<br/>


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