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

<b>let </b><font color="Maroon">S</font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E1:36_1_1"/>
<font color="Maroon">S</font> is <a href="funct_1.html#V3">constant</a>
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:36_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:36_1_1"/><i><font color="Green">E49</font></i>: 
 <a href="kurato_2.html#K5" target="_self">lim_sup</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:36_1_1"/><i><font color="Green">E54</font></i>: 
 <a href="kurato_2.html#K4" target="_self">lim_inf</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, </i>;<br/>
<div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:36_1_1_1"/>
( <font color="Maroon">S</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">S</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:36_1_1_1"/>
<font color="Maroon">S</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:36_1_1_2"/>
( <font color="Maroon">S</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">S</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:36_1_1_2"/>
<font color="Maroon">S</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:36_1_1"/>
( <font color="Maroon">S</font> is <a href="kurato_2.html#V3" target="_self">convergent</a> &amp; <font color="Maroon">S</font> is <a href="kurato_2.html#V2" target="_self">ascending</a> &amp; <font color="Maroon">S</font> is <a href="kurato_2.html#V1" target="_self">descending</a> )
 <b>by </b><i>, , , , , </i>;<br/>


</div>
