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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="E1:102"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a>
 ;<br/>

<a NAME="E2:102"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">seq</font> is <a href="seq_2.html#V2">bounded_below</a> &amp; <font color="Maroon">seq</font> is <a href="seq_2.html#V1">bounded_above</a> &amp;  <a href="partfun3.html#K6">-</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a> )
 
<b>by </b><i><a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a>, <a class="ref" href="seqm_3.html#T70">SEQM_3:70</a></i>;<br/>
<a NAME="E3:102"/>
(  <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#T32" target="_self">Th32</a>, <a class="ref" href="rinfsup1.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E4:102"/><b>then </b><i><font color="Green">E40</font></i>: 
(  <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> &amp;  <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#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="seq_2.html#D5">SEQ_2:def 5</a></i>;<br/>
<i><font color="Green">E41</font></i>:  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq</font></span>)</span> = 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K2" target="_self">inf</a> <span class="p3">(<span class="default"><a href="partfun3.html#K6">-</a> <span class="p4">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rinfsup1.html#T34" target="_self">Th34</a>, </i>
<br/>.= 
 <a href="rinfsup1.html#K2" target="_self">inf</a> <span class="p1">(<span class="default"><a href="partfun3.html#K6">-</a> <span class="p2">(<span class="default"><a href="partfun3.html#K6">-</a> <span class="p3">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <span class="p4">(<span class="default"><a href="partfun3.html#K6">-</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rinfsup1.html#T33" target="_self">Th33</a>, </i>
<br/>.= 
 <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p1">(<span class="default"><a href="partfun3.html#K6">-</a> <font color="Maroon">seq</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T36">SEQ_1:36</a></i>
;<br/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq</font></span>)</span> = 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p3">(<span class="default"><a href="partfun3.html#K6">-</a> <span class="p4">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rinfsup1.html#T34" target="_self">Th34</a>, </i>
<br/>.= 
 <a href="rinfsup1.html#K1" target="_self">sup</a> <span class="p1">(<span class="default"><a href="partfun3.html#K6">-</a> <span class="p2">(<span class="default"><a href="partfun3.html#K6">-</a> <span class="p3">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <span class="p4">(<span class="default"><a href="partfun3.html#K6">-</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rinfsup1.html#T33" target="_self">Th33</a>, <a class="ref" href="rinfsup1.html#D3" target="_self">Def3</a></i>
<br/>.= 
 <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p1">(<span class="default"><a href="partfun3.html#K6">-</a> <font color="Maroon">seq</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T36">SEQ_1:36</a></i>
;<br/>
<b>hence </b><a NAME="E7:102"/>
(  <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p1">(<span class="default"><a href="partfun3.html#K6">-</a> <font color="Maroon">seq</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">seq</font></span>)</span> &amp;  <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p1">(<span class="default"><a href="partfun3.html#K6">-</a> <font color="Maroon">seq</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">seq</font></span>)</span> )
 <b>by </b><i><a class="ref" href="rinfsup1.html#T35" target="_self">Th35</a></i>;<br/>


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