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

<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/>



<a NAME="E1:105"/><i><font color="Green">E83</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <font color="Olive">b<sub>1</sub></font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="partfun3.html#V4">nonnegative</a> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="partfun3.html#V4">nonnegative</a> holds <br/><span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:105_1"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="partfun3.html#V4">nonnegative</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="partfun3.html#V4">nonnegative</a> )
 ;<br/>

<a NAME="E2:105_1"/><b>then </b><i><font color="Green">E85</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V2">bounded_below</a> &amp; <font color="Maroon">c<sub>4</sub></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/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<a NAME="E3:105_1"/><i><font color="Green">E86</font></i>: 
(  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>3</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>4</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>3</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>4</sub></font> is <a href="seq_2.html#V4">convergent</a> )
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:105_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="rinfsup1.html#T58" target="_self">Th58</a>, <a class="ref" href="rinfsup1.html#T59" target="_self">Th59</a></i>;<br/>
<a NAME="E4:105_1"/><b>then </b><i><font color="Green">E87</font></i>: 
<span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun3.html#K5">(#)</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#T28">SEQ_2:28</a></i>;<br/>
<a NAME="E5:105_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">c<sub>4</sub></font> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:105_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="seqm_3.html#T72">SEQM_3:72</a></i>;<br/>
<a NAME="E6:105_1"/><b>then </b><i><font color="Green">E89</font></i>: 
(  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="seq_2.html#V3">bounded</a> &amp;  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="seq_2.html#V4">convergent</a> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T58" target="_self">Th58</a>, <a class="ref" href="rinfsup1.html#T59" target="_self">Th59</a></i>;<br/>
<a NAME="E7:105_1"/>
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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun3.html#K5">(#)</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:105_1_1"/>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:105_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="rinfsup1.html#E2:105_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="rinfsup1.html#T81" target="_self">Th81</a></i>;<br/>
<b>hence </b><a NAME="E2:105_1_1"/>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun3.html#K5">(#)</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:105_1"/><b>then </b><i><font color="Green">E90</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun3.html#K5">(#)</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E6:105_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="rinfsup1.html#E4:105_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="seq_2.html#T32">SEQ_2:32</a></i>;<br/>
<a NAME="E9:105_1"/><i><font color="Green">E91</font></i>: 
(  <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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> &amp;  <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>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <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>3</sub></font></span>)</span> &amp;  <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>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <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>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:105_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="rinfsup1.html#E5:105_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="rinfsup1.html#T59" target="_self">Th59</a></i>;<br/>
<b>thus </b><a NAME="E10:105_1"/>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rinfsup1.html#E9:105_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="rinfsup1.html#E8:105_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="rinfsup1.html#E3:105_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="seq_2.html#T29">SEQ_2:29</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:105"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <font color="Olive">b<sub>1</sub></font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="partfun3.html#V4">nonnegative</a> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="partfun3.html#V4">nonnegative</a> holds <br/> <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:105_2"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="partfun3.html#V4">nonnegative</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="partfun3.html#V4">nonnegative</a> )
 ;<br/>

<a NAME="E2:105_2"/><b>then </b><i><font color="Green">E85</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V1">bounded_above</a> &amp; <font color="Maroon">c<sub>4</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/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<a NAME="E3:105_2"/><i><font color="Green">E86</font></i>: 
(  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>3</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>4</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>3</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>4</sub></font> is <a href="seq_2.html#V4">convergent</a> )
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:105_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="rinfsup1.html#T58" target="_self">Th58</a>, <a class="ref" href="rinfsup1.html#T60" target="_self">Th60</a></i>;<br/>
<a NAME="E4:105_2"/><b>then </b><i><font color="Green">E87</font></i>: 
<span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun3.html#K5">(#)</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#T28">SEQ_2:28</a></i>;<br/>
<a NAME="E5:105_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">c<sub>4</sub></font> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:105_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="seqm_3.html#T72">SEQM_3:72</a></i>;<br/>
<a NAME="E6:105_2"/><b>then </b><i><font color="Green">E89</font></i>: 
(  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="seq_2.html#V3">bounded</a> &amp;  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="seq_2.html#V4">convergent</a> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T58" target="_self">Th58</a>, <a class="ref" href="rinfsup1.html#T60" target="_self">Th60</a></i>;<br/>
<a NAME="E7:105_2"/>
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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></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"><span class="p2">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun3.html#K5">(#)</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:105_2_1"/>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:105_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="rinfsup1.html#E2:105_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="rinfsup1.html#T82" target="_self">Th82</a></i>;<br/>
<b>hence </b><a NAME="E2:105_2_1"/>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun3.html#K5">(#)</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:105_2"/><b>then </b><i><font color="Green">E90</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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun3.html#K5">(#)</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E6:105_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="rinfsup1.html#E4:105_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="seq_2.html#T32">SEQ_2:32</a></i>;<br/>
<a NAME="E9:105_2"/><i><font color="Green">E91</font></i>: 
(  <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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> &amp;  <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>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <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>3</sub></font></span>)</span> &amp;  <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>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <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>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:105_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="rinfsup1.html#E5:105_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="rinfsup1.html#T60" target="_self">Th60</a></i>;<br/>
<b>thus </b><a NAME="E10:105_2"/>
 <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rinfsup1.html#E8:105_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="rinfsup1.html#E9:105_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="rinfsup1.html#E3:105_2"><i><font color="Green">E86</font></i></a>, <a class="ref" href="seq_2.html#T29">SEQ_2:29</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:105"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="partfun3.html#V4">nonnegative</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="partfun3.html#V4">nonnegative</a> implies ( <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" target="_self">lim_inf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rinfsup1.html#K6" target="_self">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" target="_self">lim_sup</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="rinfsup1.html#E1:105"><i><font color="Green">E83</font></i></a></i>;<br/>


</div>
