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<b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:65"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a>
 ;<br/>

<a NAME="E2:65"/><b>then </b>
( <font color="Maroon">seq</font> is <a href="seq_2.html#V1">bounded_above</a> &amp; <font color="Maroon">seq</font> is <a href="seq_2.html#V2">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a></i>;<br/>
<a NAME="E3:65"/><b>then </b>
(  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font> is <a href="seqm_3.html#V4">non-increasing</a> &amp;  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font> is <a href="seqm_3.html#V3">non-decreasing</a> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T8" target="_self">Th8</a>, </i>;<br/>
<a NAME="E4:65"/><b>then </b><i><font color="Green">E39</font></i>: 
(  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V1">bounded_above</a> &amp;  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V2">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="limfunc1.html#T26">LIMFUNC1:26</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:65_1"/>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K2" target="_self">inf</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span>
 <b>by </b><i><a class="ref" href="rinfsup1.html#T13" target="_self">Th13</a>, <a class="ref" href="rinfsup1.html#T9" target="_self">Th9</a></i>;<br/><b>hence </b><a NAME="E2:65_1"/>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" target="_self">inf</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E6:65"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#D3">SEQ_2:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:65_2"/>
 <a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="rinfsup1.html#T13" target="_self">Th13</a>, <a class="ref" href="rinfsup1.html#T10" target="_self">Th10</a></i>;<br/><b>hence </b><a NAME="E2:65_2"/>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E8:65"/><b>then </b>
 <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V2">bounded_below</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#D4">SEQ_2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E9:65"/>
(  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a> &amp;  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a> )
 <b>by </b><i><a class="ref" href="rinfsup1.html#T14" target="_self">Th14</a>, <a class="ref" href="rinfsup1.html#T15" target="_self">Th15</a>, <a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a></i>;<br/>


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