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<b>let </b><font color="Maroon" title="c1">seq</font> be   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a>;<br/>

<b>assume </b><a NAME="E1:55"/><i><font color="Green" title="E52">R1</font></i>: 
(  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c1">seq</font> &amp;  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 ;<br/>

<b>set </b><font color="Maroon" title="c2">a</font> =  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq</font>;<br/>
<b>consider </b><font color="Maroon" title="c3">k</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> <b> such that </b><br/><a NAME="E3:55"/><i><font color="Green" title="E53">C1</font></i>: 
<font color="Maroon" title="c1">seq</font> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c3">k</font> is <a href="rinfsup2.html#V3" title="RINFSUP2:attr.3">bounded</a>
 <b>by </b><i><a class="ref" href="rinfsup2.html#T18" target="_self" title="RINFSUP2:th.18">Lm3002q</a>, <a class="txt" href="rinfsup2.html#E1:55"><i><font color="Green" title="E52">R1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">rseq0</font> = <font color="Maroon" title="c1">seq</font> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c3">k</font> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> <b>by </b><i><a class="ref" href="rinfsup2.html#T11" target="_self" title="RINFSUP2:th.11">Lm300a</a>, <a class="txt" href="rinfsup2.html#E3:55"><i><font color="Green" title="E53">C1</font></i></a></i>;<br/>
<a NAME="E5:55"/>
( <font color="Maroon" title="c1">seq</font> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c3">k</font> is <a href="rinfsup2.html#V2" title="RINFSUP2:attr.2">bounded_above</a> &amp; <font color="Maroon" title="c1">seq</font> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c3">k</font> is <a href="rinfsup2.html#V1" title="RINFSUP2:attr.1">bounded_below</a> )
 
<b>by </b><i><a class="txt" href="rinfsup2.html#E3:55"><i><font color="Green" title="E53">C1</font></i></a>, <a class="ref" href="rinfsup2.html#D5" target="_self" title="RINFSUP2:def.5">defbounded</a></i>;<br/>
<a NAME="E6:55"/><b>then </b>
( <font color="Maroon" title="c4">rseq0</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> &amp; <font color="Maroon" title="c4">rseq0</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="rinfsup2.html#T12" target="_self" title="RINFSUP2:th.12">Lm300b</a>, <a class="ref" href="rinfsup2.html#T13" target="_self" title="RINFSUP2:th.13">Lm300c</a></i>;<br/>
<a NAME="E7:55"/><b>then </b><i><font color="Green" title="E54">C2</font></i>: 
<font color="Maroon" title="c4">rseq0</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</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="E8:55"/><b>then </b>
(  <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c4">rseq0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> &amp;  <a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c4">rseq0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="rinfsup2.html#T10" target="_self" title="RINFSUP2:th.10">Lm203</a>, <a class="ref" href="rinfsup2.html#T9" target="_self" title="RINFSUP2:th.9">Lm202</a></i>;<br/>
<a NAME="E9:55"/><b>then </b><i><font color="Green" title="E55">C3</font></i>: 
(  <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c4">rseq0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq</font> &amp;  <a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c4">rseq0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c1">seq</font> )
 
<b>by </b><i><a class="ref" href="rinfsup2.html#T28" target="_self" title="RINFSUP2:th.28">Lm203a</a>, <a class="ref" href="rinfsup2.html#T29" target="_self" title="RINFSUP2:th.29">Lm203b</a></i>;<br/>
<a NAME="E10:55"/><b>then </b>
<font color="Maroon" title="c4">rseq0</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#T90" title="RINFSUP1:th.90">RINFSUP1:90</a>, <a class="txt" href="rinfsup2.html#E7:55"><i><font color="Green" title="E54">C2</font></i></a>, <a class="txt" href="rinfsup2.html#E1:55"><i><font color="Green" title="E52">R1</font></i></a></i>;<br/>
<a NAME="E11:55"/><b>then </b>
(  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c4">rseq0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c1">seq</font> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c4">rseq0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq</font> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c4">rseq0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> &amp; <font color="Maroon" title="c1">seq</font> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c3">k</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> )
 
<b>by </b><i><a class="ref" href="rinfsup2.html#T14" target="_self" title="RINFSUP2:th.14">Lm300f</a>, <a class="ref" href="rinfsup1.html#T91" title="RINFSUP1:th.91">RINFSUP1:91</a>, <a class="txt" href="rinfsup2.html#E9:55"><i><font color="Green" title="E55">C3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:55"/>
( <font color="Maroon" title="c1">seq</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c1">seq</font> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c1">seq</font> )
 <b>by </b><i><a class="ref" href="rinfsup2.html#T17" target="_self" title="RINFSUP2:th.17">Lm3000</a></i>;<br/>


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