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<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/>



<b>assume </b><a NAME="E1:103"/><i><font color="Green" title="E83">A1</font></i>: 
( <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="c2">seq2</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp; ( 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  <font color="Maroon" title="c1">seq1</font> <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> <font color="Maroon" title="c2">seq2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> ) )
 ;<br/>

<a NAME="E2:103"/><b>then </b><i><font color="Green" title="E84">A2</font></i>: 
(  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c1">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="c1">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="c2">seq2</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="c2">seq2</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T60" target="_self" title="RINFSUP1:th.60">Th60</a>, <a class="ref" href="rinfsup1.html#T59" target="_self" title="RINFSUP1:th.59">Th59</a></i>;<br/>
<a NAME="E3:103"/><i><font color="Green" title="E85">A3</font></i>: 
( ( 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> <font color="Maroon" title="c1">seq1</font></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"><a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> ) &amp; ( 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> <font color="Maroon" title="c1">seq1</font></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"><a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> ) )
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:103"><i><font color="Green" title="E83">A1</font></i></a>, <a class="ref" href="rinfsup1.html#T77" target="_self" title="RINFSUP1:th.77">Th77</a></i>;<br/>
<a NAME="E4:103"/>
( <font color="Maroon" title="c1">seq1</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> &amp; <font color="Maroon" title="c1">seq1</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> &amp; <font color="Maroon" title="c2">seq2</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> &amp; <font color="Maroon" title="c2">seq2</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> )
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E1:103"><i><font color="Green" title="E83">A1</font></i></a>, <a class="ref" href="seq_2.html#D5" title="SEQ_2:def.5">SEQ_2:def 5</a></i>;<br/>
<a NAME="E5:103"/><b>then </b><i><font color="Green" title="E86">A4</font></i>: 
(  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq1</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a> &amp;  <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq1</font> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a> &amp;  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c2">seq2</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a> &amp;  <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">seq2</font> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a> )
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T53" target="_self" title="RINFSUP1:th.53">Th53</a>, <a class="ref" href="rinfsup1.html#T52" target="_self" title="RINFSUP1:th.52">Th52</a></i>;<br/>
<a NAME="E6:103"/>
(  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c1">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="c1">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="c2">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="c2">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#E1:103"><i><font color="Green" title="E83">A1</font></i></a>, <a class="ref" href="rinfsup1.html#T58" target="_self" title="RINFSUP1:th.58">Th58</a></i>;<br/>
<a NAME="E7:103"/><b>then </b>
(  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c1">seq1</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> &amp;  <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c1">seq1</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> &amp;  <a href="rinfsup1.html#K4" title="RINFSUP1:func.4">superior_realsequence</a> <font color="Maroon" title="c2">seq2</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> &amp;  <a href="rinfsup1.html#K3" title="RINFSUP1:func.3">inferior_realsequence</a> <font color="Maroon" title="c2">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/>
<a NAME="E8:103"/><b>then </b>
(  <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="c1">seq1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c1">seq1</font></span>)</span> &amp;  <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="c1">seq1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c1">seq1</font></span>)</span> &amp;  <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="c2">seq2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c2">seq2</font></span>)</span> &amp;  <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="c2">seq2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c2">seq2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rinfsup1.html#E5:103"><i><font color="Green" title="E86">A4</font></i></a>, <a class="ref" href="rinfsup1.html#T24" target="_self" title="RINFSUP1:th.24">Th24</a>, <a class="ref" href="rinfsup1.html#T25" target="_self" title="RINFSUP1:th.25">Th25</a></i>;<br/>
<b>hence </b><a NAME="E9:103"/>
(  <a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c1">seq1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c2">seq2</font> &amp;  <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c1">seq1</font> <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> <font color="Maroon" title="c2">seq2</font> )
 <b>by </b><i><a class="txt" href="rinfsup1.html#E2:103"><i><font color="Green" title="E84">A2</font></i></a>, <a class="txt" href="rinfsup1.html#E3:103"><i><font color="Green" title="E85">A3</font></i></a>, <a class="ref" href="seq_2.html#T32" title="SEQ_2:th.32">SEQ_2:32</a></i>;<br/>


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