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<span class="kw">let </span><font color="Maroon" title="c1">seq</font> be   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  implies ( <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> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:51"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E2:51"/><span class="lab"><font color="Green" title="E46">A2</font></span>: 
 <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="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">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> <span class="kw"> such that </span><br/><a NAME="E4:51"/><span class="lab"><font color="Green" title="E47">A3</font></span>: 
<font color="Maroon" title="c1">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c2">k</font> is  <a href="rinfsup2.html#V3" title="RINFSUP2:attr.3">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="ref" href="rinfsup2.html#T18" target="_self" title="RINFSUP2:th.18">Th18</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">rseq0</font> = <font color="Maroon" title="c1">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c2">k</font> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="rinfsup2.html#T11" target="_self" title="RINFSUP2:th.11">Th11</a></span>;<br/>
<a NAME="E6:51"/>
<font color="Maroon" title="c1">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c2">k</font> is  <a href="rinfsup2.html#V1" title="RINFSUP2:attr.1">bounded_below</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="rinfsup2.html#D5" target="_self" title="RINFSUP2:def.5">Def5</a></span>;<br/>
<a NAME="E7:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A4</font></span>: 
<font color="Maroon" title="c3">rseq0</font> is  <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#T13" target="_self" title="RINFSUP2:th.13">Th13</a></span>;<br/>
<a NAME="E8:51"/>
<font color="Maroon" title="c1">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c2">k</font> is  <a href="rinfsup2.html#V2" title="RINFSUP2:attr.2">bounded_above</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="rinfsup2.html#D5" target="_self" title="RINFSUP2:def.5">Def5</a></span>;<br/>
<a NAME="E9:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A5</font></span>: 
<font color="Maroon" title="c3">rseq0</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#T12" target="_self" title="RINFSUP2:th.12">Th12</a></span>;<br/>
<a NAME="E10:51"/><span class="kw">then </span>
 <a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c3">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="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c2">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:51"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="ref" href="rinfsup2.html#T9" target="_self" title="RINFSUP2:th.9">Th9</a></span>;<br/>
<a NAME="E11:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A6</font></span>: 
 <a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c3">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#T28" target="_self" title="RINFSUP2:th.28">Th28</a></span>;<br/>
<a NAME="E12:51"/>
 <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c3">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="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c2">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:51"><span class="lab"><font color="Green" title="E49">A5</font></span></a>, <a class="txt" href="#E7:51"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="ref" href="rinfsup2.html#T10" target="_self" title="RINFSUP2:th.10">Th10</a></span>;<br/>
<a NAME="E13:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A7</font></span>: 
 <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <font color="Maroon" title="c3">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#T29" target="_self" title="RINFSUP2:th.29">Th29</a></span>;<br/>
<a NAME="E14:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A8</font></span>: 
<font color="Maroon" title="c3">rseq0</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E9:51"><span class="lab"><font color="Green" title="E49">A5</font></span></a>, <a class="txt" href="#E7:51"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="txt" href="#E11:51"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="ref" href="rinfsup1.html#T88" title="RINFSUP1:th.88">RINFSUP1:88</a></span>;<br/>
<a NAME="E15:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A9</font></span>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:51"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="ref" href="rinfsup1.html#T89" title="RINFSUP1:th.89">RINFSUP1:89</a></span>;<br/>
<a NAME="E16:51"/><span class="lab"><font color="Green" title="E54">A10</font></span>: 
<font color="Maroon" title="c1">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c2">k</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51"><span class="lab"><font color="Green" title="E52">A8</font></span></a>, <a class="ref" href="rinfsup2.html#T14" target="_self" title="RINFSUP2:th.14">Th14</a></span>;<br/>
<a NAME="E17:51"/><span class="lab"><font color="Green" title="E55">A11</font></span>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">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="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c2">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51"><span class="lab"><font color="Green" title="E52">A8</font></span></a>, <a class="ref" href="rinfsup2.html#T14" target="_self" title="RINFSUP2:th.14">Th14</a></span>;<br/>
<a NAME="E18:51"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:51"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="txt" href="#E14:51"><span class="lab"><font color="Green" title="E52">A8</font></span></a>, <a class="ref" href="rinfsup1.html#T89" title="RINFSUP1:th.89">RINFSUP1:89</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:51"/>
( <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:51"><span class="lab"><font color="Green" title="E53">A9</font></span></a>, <a class="txt" href="#E17:51"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="txt" href="#E16:51"><span class="lab"><font color="Green" title="E54">A10</font></span></a>, <a class="ref" href="rinfsup2.html#T17" target="_self" title="RINFSUP2:th.17">Th17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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