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

<b>let </b><font color="Maroon" title="c1">seq1</font>, <font color="Maroon" title="c2">seq2</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>



<a NAME="E1:105"/><i><font color="Green" title="E83">A1</font></i>: 
for <font color="Olive" title="b1">seq1</font>, <font color="Olive" title="b2">seq2</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st <font color="Olive" title="b1">seq1</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp; <font color="Olive" title="b1">seq1</font> is <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</a> &amp; <font color="Olive" title="b2">seq2</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp; <font color="Olive" title="b2">seq2</font> is <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</a> holds <br/><span class="p1">(<span class="default"><a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Olive" title="b1">seq1</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Olive" title="b2">seq2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">seq1</font> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <font color="Olive" title="b2">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" title="c3">seq1</font>, <font color="Maroon" title="c4">seq2</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:105_1"/><i><font color="Green" title="E84">A2</font></i>: 
( <font color="Maroon" title="c3">seq1</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp; <font color="Maroon" title="c3">seq1</font> is <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</a> &amp; <font color="Maroon" title="c4">seq2</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp; <font color="Maroon" title="c4">seq2</font> is <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</a> )
 ;<br/>

<a NAME="E2:105_1"/><b>then </b><i><font color="Green" title="E85">A3</font></i>: 
( <font color="Maroon" title="c3">seq1</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> &amp; <font color="Maroon" title="c4">seq2</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="seq_2.html#D5" title="SEQ_2:def.5">SEQ_2:def 5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">seq3</font> =  <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">seq1</font>;<br/>
<b>set </b><font color="Maroon" title="c6">seq4</font> =  <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c4">seq2</font>;<br/>
<b>set </b><font color="Maroon" title="c7">seq5</font> =  <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span>;<br/>
<a NAME="E3:105_1"/><i><font color="Green" title="E86">A4</font></i>: 
(  <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">seq1</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp;  <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c4">seq2</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp;  <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">seq1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c4">seq2</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> )
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:105_1"><i><font color="Green" title="E84">A2</font></i></a>, <a class="ref" href="rinfsup1.html#T58" target="_self" title="RINFSUP1:th.58">Th58</a>, <a class="ref" href="rinfsup1.html#T59" target="_self" title="RINFSUP1:th.59">Th59</a></i>;<br/>
<a NAME="E4:105_1"/><b>then </b><i><font color="Green" title="E87">A5</font></i>: 
<span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#T28" title="SEQ_2:th.28">SEQ_2:28</a></i>;<br/>
<a NAME="E5:105_1"/><i><font color="Green" title="E88">A6</font></i>: 
<font color="Maroon" title="c3">seq1</font> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <font color="Maroon" title="c4">seq2</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:105_1"><i><font color="Green" title="E84">A2</font></i></a>, <a class="ref" href="seqm_3.html#T72" title="SEQM_3:th.72">SEQM_3:72</a></i>;<br/>
<a NAME="E6:105_1"/><b>then </b><i><font color="Green" title="E89">A7</font></i>: 
(  <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp;  <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T58" target="_self" title="RINFSUP1:th.58">Th58</a>, <a class="ref" href="rinfsup1.html#T59" target="_self" title="RINFSUP1:th.59">Th59</a></i>;<br/>
<a NAME="E7:105_1"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">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" title="c8">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:105_1_1"/>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:105_1"><i><font color="Green" title="E84">A2</font></i></a>, <a class="txt" href="rinfsup1.html#E2:105_1"><i><font color="Green" title="E85">A3</font></i></a>, <a class="ref" href="rinfsup1.html#T81" target="_self" title="RINFSUP1:th.81">Th81</a></i>;<br/>
<b>hence </b><a NAME="E2:105_1_1"/>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:105_1"/><b>then </b><i><font color="Green" title="E90">A8</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E6:105_1"><i><font color="Green" title="E89">A7</font></i></a>, <a class="txt" href="rinfsup1.html#E4:105_1"><i><font color="Green" title="E87">A5</font></i></a>, <a class="ref" href="seq_2.html#T32" title="SEQ_2:th.32">SEQ_2:32</a></i>;<br/>
<a NAME="E9:105_1"/>
(  <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> &amp;  <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> &amp;  <a href="rinfsup1.html#K1" title="RINFSUP1:func.1">sup</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:105_1"><i><font color="Green" title="E84">A2</font></i></a>, <a class="txt" href="rinfsup1.html#E5:105_1"><i><font color="Green" title="E88">A6</font></i></a>, <a class="ref" href="rinfsup1.html#T59" target="_self" title="RINFSUP1:th.59">Th59</a></i>;<br/>
<b>hence </b><a NAME="E10:105_1"/>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c4">seq2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span>
 <b>by </b><i><a class="txt" href="rinfsup1.html#E8:105_1"><i><font color="Green" title="E90">A8</font></i></a>, <a class="txt" href="rinfsup1.html#E3:105_1"><i><font color="Green" title="E86">A4</font></i></a>, <a class="ref" href="seq_2.html#T29" title="SEQ_2:th.29">SEQ_2:29</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:105"/>
for <font color="Olive" title="b1">seq1</font>, <font color="Olive" title="b2">seq2</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st <font color="Olive" title="b1">seq1</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp; <font color="Olive" title="b1">seq1</font> is <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</a> &amp; <font color="Olive" title="b2">seq2</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp; <font color="Olive" title="b2">seq2</font> is <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</a> holds <br/> <a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">seq1</font> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <font color="Olive" title="b2">seq2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Olive" title="b1">seq1</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Olive" title="b2">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" title="c3">seq1</font>, <font color="Maroon" title="c4">seq2</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:105_2"/><i><font color="Green" title="E84">A9</font></i>: 
( <font color="Maroon" title="c3">seq1</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp; <font color="Maroon" title="c3">seq1</font> is <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</a> &amp; <font color="Maroon" title="c4">seq2</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp; <font color="Maroon" title="c4">seq2</font> is <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</a> )
 ;<br/>

<a NAME="E2:105_2"/><b>then </b><i><font color="Green" title="E85">A10</font></i>: 
( <font color="Maroon" title="c3">seq1</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> &amp; <font color="Maroon" title="c4">seq2</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> )
 
<b>by </b><i><a class="ref" href="seq_2.html#D5" title="SEQ_2:def.5">SEQ_2:def 5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">seq3</font> =  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c3">seq1</font>;<br/>
<b>set </b><font color="Maroon" title="c6">seq4</font> =  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c4">seq2</font>;<br/>
<b>set </b><font color="Maroon" title="c7">seq5</font> =  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span>;<br/>
<a NAME="E3:105_2"/><i><font color="Green" title="E86">A11</font></i>: 
(  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c3">seq1</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp;  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c4">seq2</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp;  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c3">seq1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c4">seq2</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> )
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:105_2"><i><font color="Green" title="E84">A9</font></i></a>, <a class="ref" href="rinfsup1.html#T58" target="_self" title="RINFSUP1:th.58">Th58</a>, <a class="ref" href="rinfsup1.html#T60" target="_self" title="RINFSUP1:th.60">Th60</a></i>;<br/>
<a NAME="E4:105_2"/><b>then </b><i><font color="Green" title="E87">A12</font></i>: 
<span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#T28" title="SEQ_2:th.28">SEQ_2:28</a></i>;<br/>
<a NAME="E5:105_2"/><i><font color="Green" title="E88">A13</font></i>: 
<font color="Maroon" title="c3">seq1</font> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <font color="Maroon" title="c4">seq2</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:105_2"><i><font color="Green" title="E84">A9</font></i></a>, <a class="ref" href="seqm_3.html#T72" title="SEQM_3:th.72">SEQM_3:72</a></i>;<br/>
<a NAME="E6:105_2"/><b>then </b><i><font color="Green" title="E89">A14</font></i>: 
(  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp;  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T58" target="_self" title="RINFSUP1:th.58">Th58</a>, <a class="ref" href="rinfsup1.html#T60" target="_self" title="RINFSUP1:th.60">Th60</a></i>;<br/>
<a NAME="E7:105_2"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">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" title="c8">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:105_2_1"/>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:105_2"><i><font color="Green" title="E84">A9</font></i></a>, <a class="txt" href="rinfsup1.html#E2:105_2"><i><font color="Green" title="E85">A10</font></i></a>, <a class="ref" href="rinfsup1.html#T82" target="_self" title="RINFSUP1:th.82">Th82</a></i>;<br/>
<b>hence </b><a NAME="E2:105_2_1"/>
<span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:105_2"/><b>then </b><i><font color="Green" title="E90">A15</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <span class="p2">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E6:105_2"><i><font color="Green" title="E89">A14</font></i></a>, <a class="txt" href="rinfsup1.html#E4:105_2"><i><font color="Green" title="E87">A12</font></i></a>, <a class="ref" href="seq_2.html#T32" title="SEQ_2:th.32">SEQ_2:32</a></i>;<br/>
<a NAME="E9:105_2"/>
(  <a href="rinfsup1.html#K2" title="RINFSUP1:func.2">inf</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span> &amp;  <a href="rinfsup1.html#K2" title="RINFSUP1:func.2">inf</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c3">seq1</font></span>)</span> &amp;  <a href="rinfsup1.html#K2" title="RINFSUP1:func.2">inf</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c4">seq2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:105_2"><i><font color="Green" title="E84">A9</font></i></a>, <a class="txt" href="rinfsup1.html#E5:105_2"><i><font color="Green" title="E88">A13</font></i></a>, <a class="ref" href="rinfsup1.html#T60" target="_self" title="RINFSUP1:th.60">Th60</a></i>;<br/>
<b>hence </b><a NAME="E10:105_2"/>
 <a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <font color="Maroon" title="c4">seq2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c4">seq2</font></span>)</span>
 <b>by </b><i><a class="txt" href="rinfsup1.html#E8:105_2"><i><font color="Green" title="E90">A15</font></i></a>, <a class="txt" href="rinfsup1.html#E3:105_2"><i><font color="Green" title="E86">A11</font></i></a>, <a class="ref" href="seq_2.html#T29" title="SEQ_2:th.29">SEQ_2:29</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:105"/>
( <font color="Maroon" title="c1">seq1</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp; <font color="Maroon" title="c1">seq1</font> is <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</a> &amp; <font color="Maroon" title="c2">seq2</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp; <font color="Maroon" title="c2">seq2</font> is <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</a> implies ( <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> &amp;  <a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="partfun3.html#K5" title="PARTFUN3:func.5">(#)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c2">seq2</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="rinfsup1.html#E1:105"><i><font color="Green" title="E83">A1</font></i></a></i>;<br/>


</div>
