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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="Red">::  thesis: </font></span></a><span class="hide"> ( <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="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a>  implies ( <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> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
<font color="Maroon" title="c1">seq</font> is  <a href="rinfsup2.html#V3" title="RINFSUP2:attr.3">bounded</a> 
 <span class="kw">and </span><br/><a NAME="E2:40"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
<font color="Maroon" title="c1">seq</font> is  <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> )</span><br/>

<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="rinfsup2.html#T11" target="_self" title="RINFSUP2:th.11">Th11</a></span>;<br/>
<a NAME="E4:40"/><span class="lab"><font color="Green" title="E37">A3</font></span>: 
<font color="Maroon" title="c1">seq</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="#E1:40"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="rinfsup2.html#D5" target="_self" title="RINFSUP2:def.5">Def5</a></span>;<br/>
<a NAME="E5:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A4</font></span>: 
<font color="Maroon" title="c2">rseq</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="E6:40"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="rinfsup1.html#T91" title="RINFSUP1:th.91">RINFSUP1:91</a></span>;<br/>
<a NAME="E7:40"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="rinfsup1.html#T66" title="RINFSUP1:th.66">RINFSUP1:66</a></span>;<br/>
<a NAME="E8:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A5</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="ref" href="rinfsup2.html#T14" target="_self" title="RINFSUP2:th.14">Th14</a></span>;<br/>
<a NAME="E9:40"/>
 <a href="supinf_2.html#K18" title="SUPINF_2:func.18">rng</a> <font color="Maroon" title="c1">seq</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="ref" href="rinfsup2.html#D4" target="_self" title="RINFSUP2:def.4">Def4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:40"/>
( <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="txt" href="#E8:40"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="ref" href="rinfsup2.html#T1" target="_self" title="RINFSUP2:th.1">Th1</a>, <a class="ref" href="rinfsup2.html#T14" target="_self" title="RINFSUP2:th.14">Th14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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