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

<b>let </b><font color="Maroon">seq1</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">seq2</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<a NAME="E1:105"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">seq1</font>, <font color="Olive">seq2</font> being  <a href="seq_1.html#NM1">Real_Sequence</a>  st <font color="Olive">seq1</font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Olive">seq1</font> is <a href="partfun3.html#V4">nonnegative</a> &amp; <font color="Olive">seq2</font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Olive">seq2</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">seq1</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">seq2</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">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Olive">seq2</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">seq1</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">seq2</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:105_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">seq1</font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Maroon">seq1</font> is <a href="partfun3.html#V4">nonnegative</a> &amp; <font color="Maroon">seq2</font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Maroon">seq2</font> is <a href="partfun3.html#V4">nonnegative</a> )
 ;<br/>

<a NAME="E2:105_1"/><b>then </b><i><font color="Green">E40</font></i>: 
( <font color="Maroon">seq1</font> is <a href="seq_2.html#V2">bounded_below</a> &amp; <font color="Maroon">seq2</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">seq3</font> =  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq1</font>;<br/>
<b>set </b><font color="Maroon">seq4</font> =  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq2</font>;<br/>
<b>set </b><font color="Maroon">seq5</font> =  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</font></span>)</span>;<br/>
<a NAME="E3:105_1"/><i><font color="Green">E41</font></i>: 
(  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq1</font> is <a href="seq_2.html#V3">bounded</a> &amp;  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq2</font> is <a href="seq_2.html#V3">bounded</a> &amp;  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq1</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq2</font> is <a href="seq_2.html#V4">convergent</a> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T33" target="_self">Th33</a>, <a class="ref" href="rinfsup1.html#T12" target="_self">Th12</a>, <a class="ref" href="rinfsup1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E4:105_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <font color="Maroon">seq1</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">seq2</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">E50</font></i>: 
<font color="Maroon">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</font> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T33" target="_self">Th33</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">E51</font></i>: 
(  <a href="rinfsup1.html#K3" target="_self">inferior_realsequence</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</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">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</font></span>)</span> is <a href="seq_2.html#V4">convergent</a> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T12" target="_self">Th12</a>, <a class="ref" href="rinfsup1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E7:105_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</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">seq1</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">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</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">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</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">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T33" target="_self">Th33</a>, <a class="ref" href="rinfsup1.html#T34" target="_self">Th34</a>, <a class="ref" href="rinfsup1.html#T27" target="_self">Th27</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">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</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">seq1</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">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</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">E52</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">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</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">seq1</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">seq2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T37" target="_self">Th37</a>, <a class="ref" href="rinfsup1.html#T36" target="_self">Th36</a>, <a class="ref" href="seq_2.html#T32">SEQ_2:32</a></i>;<br/>
<a NAME="E9:105_1"/><i><font color="Green">E53</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">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</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">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</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">seq1</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">seq1</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">seq2</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">seq2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T33" target="_self">Th33</a>, , <a class="ref" href="rinfsup1.html#T13" target="_self">Th13</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">seq1</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">seq2</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">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</font></span>)</span>
 <b>by </b><i><a class="ref" href="rinfsup1.html#D5" target="_self">Def5</a>, <a class="ref" href="rinfsup1.html#D4" target="_self">Def4</a>, <a class="ref" href="rinfsup1.html#T35" target="_self">Th35</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">seq1</font>, <font color="Olive">seq2</font> being  <a href="seq_1.html#NM1">Real_Sequence</a>  st <font color="Olive">seq1</font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Olive">seq1</font> is <a href="partfun3.html#V4">nonnegative</a> &amp; <font color="Olive">seq2</font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Olive">seq2</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">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Olive">seq2</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">seq1</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">seq2</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">seq1</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">seq2</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:105_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">seq1</font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Maroon">seq1</font> is <a href="partfun3.html#V4">nonnegative</a> &amp; <font color="Maroon">seq2</font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Maroon">seq2</font> is <a href="partfun3.html#V4">nonnegative</a> )
 ;<br/>

<a NAME="E2:105_2"/><b>then </b><i><font color="Green">E98</font></i>: 
( <font color="Maroon">seq1</font> is <a href="seq_2.html#V1">bounded_above</a> &amp; <font color="Maroon">seq2</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">seq3</font> =  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq1</font>;<br/>
<b>set </b><font color="Maroon">seq4</font> =  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq2</font>;<br/>
<b>set </b><font color="Maroon">seq5</font> =  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</font></span>)</span>;<br/>
<a NAME="E3:105_2"/><i><font color="Green">E99</font></i>: 
(  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq1</font> is <a href="seq_2.html#V3">bounded</a> &amp;  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq2</font> is <a href="seq_2.html#V3">bounded</a> &amp;  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq1</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq2</font> is <a href="seq_2.html#V4">convergent</a> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T33" target="_self">Th33</a>, <a class="ref" href="rinfsup1.html#T12" target="_self">Th12</a>, <a class="ref" href="rinfsup1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E4:105_2"/><b>then </b><i><font color="Green">E100</font></i>: 
<span class="p1">(<span class="default"><a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <font color="Maroon">seq1</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">seq2</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">E101</font></i>: 
<font color="Maroon">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</font> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T33" target="_self">Th33</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">E102</font></i>: 
(  <a href="rinfsup1.html#K4" target="_self">superior_realsequence</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</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">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</font></span>)</span> is <a href="seq_2.html#V4">convergent</a> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T12" target="_self">Th12</a>, <a class="ref" href="rinfsup1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E7:105_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</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">seq1</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">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</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">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</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">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T33" target="_self">Th33</a>, <a class="ref" href="rinfsup1.html#T34" target="_self">Th34</a>, <a class="ref" href="rinfsup1.html#T29" target="_self">Th29</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">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</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">seq1</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">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</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">E103</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">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</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">seq1</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">seq2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T37" target="_self">Th37</a>, <a class="ref" href="rinfsup1.html#T36" target="_self">Th36</a>, <a class="ref" href="seq_2.html#T32">SEQ_2:32</a></i>;<br/>
<a NAME="E9:105_2"/><i><font color="Green">E104</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">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</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">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</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">seq1</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">seq1</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">seq2</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">seq2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T33" target="_self">Th33</a>, , <a class="ref" href="rinfsup1.html#T14" target="_self">Th14</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">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</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">seq1</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">seq2</font></span>)</span>
 <b>by </b><i><a class="ref" href="rinfsup1.html#D4" target="_self">Def4</a>, <a class="ref" href="rinfsup1.html#D5" target="_self">Def5</a>, <a class="ref" href="rinfsup1.html#T35" target="_self">Th35</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">seq1</font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Maroon">seq1</font> is <a href="partfun3.html#V4">nonnegative</a> &amp; <font color="Maroon">seq2</font> is <a href="seq_2.html#V3">bounded</a> &amp; <font color="Maroon">seq2</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">seq1</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">seq2</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">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</font></span>)</span> &amp;  <a href="rinfsup1.html#K5" target="_self">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">seq2</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">seq1</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">seq2</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="rinfsup1.html#T32" target="_self">Th32</a></i>;<br/>


</div>
