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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="E1:44"/><i><font color="Green" title="E42">A1</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#V6" title="RINFSUP2:attr.6">non-decreasing</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">Th11</a></i>;<br/>
<a NAME="E3:44"/><i><font color="Green" title="E43">A2</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="rinfsup2.html#V2" title="RINFSUP2:attr.2">bounded_above</a>
 
<b>by </b><i><a class="txt" href="rinfsup2.html#E1:44"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="rinfsup2.html#D5" target="_self" title="RINFSUP2:def.5">Def5</a></i>;<br/>
<a NAME="E4:44"/><b>then </b><i><font color="Green" title="E44">A3</font></i>: 
<font color="Maroon" title="c2">rseq</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="rinfsup2.html#T12" target="_self" title="RINFSUP2:th.12">Th12</a></i>;<br/>
<a NAME="E5:44"/><i><font color="Green" title="E45">A4</font></i>: 
<font color="Maroon" title="c2">rseq</font> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a>
 
<b>by </b><i><a class="txt" href="rinfsup2.html#E1:44"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="rinfsup2.html#E29"><i><font color="Green" title="E24">Lm3</font></i></a></i>;<br/>
<a NAME="E6:44"/><b>then </b><i><font color="Green" title="E46">A5</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#E4:44"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="seq_4.html#T51" title="SEQ_4:th.51">SEQ_4:51</a></i>;<br/>
<a NAME="E7:44"/><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#K6" title="RINFSUP1:func.6">lim_inf</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="E8:44"/><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#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c2">rseq</font>
 
<b>by </b><i><a class="txt" href="rinfsup2.html#E5:44"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="rinfsup1.html#T66" title="RINFSUP1:th.66">RINFSUP1:66</a></i>;<br/>
<a NAME="E9:44"/><b>then </b><i><font color="Green" title="E47">A6</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#K1" title="RINFSUP1:func.1">sup</a> <font color="Maroon" title="c2">rseq</font>
 
<b>by </b><i><a class="txt" href="rinfsup2.html#E6:44"><i><font color="Green" title="E46">A5</font></i></a>, <a class="ref" href="rinfsup2.html#T14" target="_self" title="RINFSUP2:th.14">Th14</a></i>;<br/>
<a NAME="E10:44"/>
 <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#V3" title="SUPINF_1:attr.3">bounded_above</a>
 
<b>by </b><i><a class="txt" href="rinfsup2.html#E3:44"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="rinfsup2.html#D4" target="_self" title="RINFSUP2:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E11:44"/>
( <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#K1" title="RINFSUP2:func.1">sup</a> <font color="Maroon" title="c1">seq</font> )
 <b>by </b><i><a class="txt" href="rinfsup2.html#E6:44"><i><font color="Green" title="E46">A5</font></i></a>, <a class="txt" href="rinfsup2.html#E9:44"><i><font color="Green" title="E47">A6</font></i></a>, <a class="ref" href="rinfsup2.html#T2" target="_self" title="RINFSUP2:th.2">Th2</a>, <a class="ref" href="rinfsup2.html#T14" target="_self" title="RINFSUP2:th.14">Th14</a></i>;<br/>


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