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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:103"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V3">bounded</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>

<a NAME="E2:103"/><b>then </b><i><font color="Green">E84</font></i>: 
(  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V4">convergent</a> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T60" target="_self">Th60</a>, <a class="ref" href="rinfsup1.html#T59" target="_self">Th59</a></i>;<br/>
<a NAME="E3:103"/><i><font color="Green">E85</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <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">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:103"><i><font color="Green">E83</font></i></a>, <a class="ref" href="rinfsup1.html#T77" target="_self">Th77</a></i>;<br/>
<a NAME="E4:103"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V1">bounded_above</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V2">bounded_below</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V1">bounded_above</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V2">bounded_below</a> )
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:103"><i><font color="Green">E83</font></i></a>, <a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a></i>;<br/>
<a NAME="E5:103"/><b>then </b><i><font color="Green">E86</font></i>: 
(  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font> is <a href="seqm_3.html#V3">non-decreasing</a> &amp;  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font> is <a href="seqm_3.html#V3">non-decreasing</a> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T53" target="_self">Th53</a>, <a class="ref" href="rinfsup1.html#T52" target="_self">Th52</a></i>;<br/>
<a NAME="E6:103"/>
(  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V3">bounded</a> &amp;  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V3">bounded</a> &amp;  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V3">bounded</a> &amp;  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V3">bounded</a> )
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:103"><i><font color="Green">E83</font></i></a>, <a class="ref" href="rinfsup1.html#T58" target="_self">Th58</a></i>;<br/>
<a NAME="E7:103"/><b>then </b>
(  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V2">bounded_below</a> &amp;  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V1">bounded_above</a> &amp;  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V2">bounded_below</a> &amp;  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V1">bounded_above</a> )
 
<b>by </b><i><a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a></i>;<br/>
<a NAME="E8:103"/><b>then </b><i><font color="Green">E87</font></i>: 
(  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</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">c<sub>1</sub></font></span>)</span> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <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">c<sub>1</sub></font></span>)</span> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</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">c<sub>2</sub></font></span>)</span> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <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">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E5:103"><i><font color="Green">E86</font></i></a>, <a class="ref" href="rinfsup1.html#T24" target="_self">Th24</a>, <a class="ref" href="rinfsup1.html#T25" target="_self">Th25</a></i>;<br/>
<b>thus </b><a NAME="E9:103"/>
(  <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="rinfsup1.html#E8:103"><i><font color="Green">E87</font></i></a>, <a class="txt" href="rinfsup1.html#E2:103"><i><font color="Green">E84</font></i></a>, <a class="txt" href="rinfsup1.html#E3:103"><i><font color="Green">E85</font></i></a>, <a class="ref" href="seq_2.html#T32">SEQ_2:32</a></i>;<br/>


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