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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="E1:43"/><i><font color="Green" title="E41">AS</font></i>: 
( <font color="Maroon" title="c1">seq</font> is <a href="rinfsup2.html#V3" title="RINFSUP2:attr.3">bounded</a> &amp; <font color="Maroon" title="c1">seq</font> is <a href="rinfsup2.html#V7" title="RINFSUP2:attr.7">non-increasing</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c2">rseq</font> = <font color="Maroon" title="c1">seq</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></i>;<br/>
<a NAME="E3:43"/><i><font color="Green" title="E42">P0</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="rinfsup2.html#V1" title="RINFSUP2:attr.1">bounded_below</a>
 
<b>by </b><i><a class="txt" href="rinfsup2.html#E1:43"><i><font color="Green" title="E41">AS</font></i></a>, <a class="ref" href="rinfsup2.html#D5" target="_self" title="RINFSUP2:def.5">defbounded</a></i>;<br/>
<a NAME="E4:43"/><b>then </b><i><font color="Green" title="E43">P2</font></i>: 
<font color="Maroon" title="c2">rseq</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#T13" target="_self" title="RINFSUP2:th.13">Lm300c</a></i>;<br/>
<a NAME="E5:43"/><i><font color="Green" title="E44">P4</font></i>: 
<font color="Maroon" title="c2">rseq</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a>
 
<b>by </b><i><a class="txt" href="rinfsup2.html#E1:43"><i><font color="Green" title="E41">AS</font></i></a>, <a class="txt" href="rinfsup2.html#E29"><i><font color="Green" title="E24">Lm300e</font></i></a></i>;<br/>
<a NAME="E6:43"/><b>then </b><i><font color="Green" title="E45">P5</font></i>: 
<font color="Maroon" title="c2">rseq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="rinfsup2.html#E4:43"><i><font color="Green" title="E43">P2</font></i></a>, <a class="ref" href="seq_4.html#T52" title="SEQ_4:th.52">SEQ_4:52</a></i>;<br/>
<a NAME="E7:43"/><b>then </b>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">rseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup1.html#K5" title="RINFSUP1:func.5">lim_sup</a> <font color="Maroon" title="c2">rseq</font>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#T91" title="RINFSUP1:th.91">RINFSUP1:91</a></i>;<br/>
<a NAME="E8:43"/><b>then </b>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">rseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup1.html#K2" title="RINFSUP1:func.2">inf</a> <font color="Maroon" title="c2">rseq</font>
 
<b>by </b><i><a class="txt" href="rinfsup2.html#E5:43"><i><font color="Green" title="E44">P4</font></i></a>, <a class="ref" href="rinfsup1.html#T72" title="RINFSUP1:th.72">RINFSUP1:72</a></i>;<br/>
<a NAME="E9:43"/><b>then </b><i><font color="Green" title="E46">P7</font></i>: 
 <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="rinfsup1.html#K2" title="RINFSUP1:func.2">inf</a> <font color="Maroon" title="c2">rseq</font>
 
<b>by </b><i><a class="ref" href="rinfsup2.html#T14" target="_self" title="RINFSUP2:th.14">Lm300f</a>, <a class="txt" href="rinfsup2.html#E6:43"><i><font color="Green" title="E45">P5</font></i></a></i>;<br/>
<a NAME="E10:43"/>
 <a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <font color="Maroon" title="c1">seq</font> is <a href="supinf_1.html#V4" title="SUPINF_1:attr.4">bounded_below</a>
 
<b>by </b><i><a class="txt" href="rinfsup2.html#E3:43"><i><font color="Green" title="E42">P0</font></i></a>, <a class="ref" href="rinfsup2.html#D3" target="_self" title="RINFSUP2:def.3">defboundedbelow</a></i>;<br/>
<b>hence </b><a NAME="E11:43"/>
( <font color="Maroon" title="c1">seq</font> is <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a> &amp; <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#K2" title="RINFSUP2:func.2">inf</a> <font color="Maroon" title="c1">seq</font> )
 <b>by </b><i><a class="ref" href="rinfsup2.html#T14" target="_self" title="RINFSUP2:th.14">Lm300f</a>, <a class="ref" href="rinfsup2.html#T4" target="_self" title="RINFSUP2:th.4">Lm022x</a>, <a class="txt" href="rinfsup2.html#E6:43"><i><font color="Green" title="E45">P5</font></i></a>, <a class="txt" href="rinfsup2.html#E9:43"><i><font color="Green" title="E46">P7</font></i></a></i>;<br/>


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