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<b>let </b><font color="Maroon" title="c1">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:45"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>
 <b>and </b><br/><a NAME="E2:45"/><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a>
 ;<br/>

<a NAME="E3:45"/><i><font color="Green" title="E35">A3</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="seq_4.html#E1:45"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="seq_4.html#E2:45"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="seq_4.html#T51" target="_self" title="SEQ_4:th.51">Th51</a></i>;<br/>
<b>let </b><font color="Maroon" title="c2">m</font> be    <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> ;<br/>

<b>reconsider </b><font color="Maroon" title="c3">seq1</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">m</font></span>)</span> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> <b>by </b><i><a class="ref" href="funcop_1.html#T57" title="FUNCOP_1:th.57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E5:45"/><i><font color="Green" title="E36">A4</font></i>: 
<font color="Maroon" title="c3">seq1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T39" target="_self" title="SEQ_4:th.39">Th39</a></i>;<br/>
<a NAME="E6:45"/>
<font color="Maroon" title="c3">seq1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E7:45"/><b>then </b><i><font color="Green" title="E37">A5</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="ref" href="seq_4.html#T40" target="_self" title="SEQ_4:th.40">Th40</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <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>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <font color="Maroon" title="c1">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">m</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c4">seq2</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><b> such that </b><br/><a NAME="E9:45"/><i><font color="Green" title="E38">A6</font></i>: 
for <font color="Olive" title="b1">n</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>  holds  <font color="Maroon" title="c4">seq2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1" title="SEQ_1:sch.1">SEQ_1:sch 1</a>();<br/></i>
<a NAME="E10:45"/><i><font color="Green" title="E39">A7</font></i>: 
<font color="Maroon" title="c4">seq2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c1">seq</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="txt" href="seq_4.html#E9:45"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>;<br/>
<a NAME="E11:45"/><b>then </b><i><font color="Green" title="E40">A8</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c4">seq2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c1">seq</font>
 
<b>by </b><i><a class="txt" href="seq_4.html#E3:45"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="seq_4.html#T33" target="_self" title="SEQ_4:th.33">Th33</a></i>;<br/>
<a NAME="E12:45"/><i><font color="Green" title="E41">A9</font></i>: 
<font color="Maroon" title="c4">seq2</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="seq_4.html#E3:45"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="seq_4.html#E10:45"><i><font color="Green" title="E39">A7</font></i></a>, <a class="ref" href="seq_4.html#T33" target="_self" title="SEQ_4:th.33">Th33</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">n</font> be    <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> ;<br/><a NAME="E1:45_1"/><i><font color="Green" title="E42">A10</font></i>: 
<font color="Maroon" title="c3">seq1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><a NAME="E2:45_1"/>
<font color="Maroon" title="c4">seq2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="seq_4.html#E9:45"><i><font color="Green" title="E38">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E3:45_1"/>
<font color="Maroon" title="c3">seq1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">seq2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E2:45"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="seq_4.html#E1:45_1"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="seqm_3.html#T11" title="SEQM_3:th.11">SEQM_3:11</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:45"/>
<font color="Maroon" title="c1">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c1">seq</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E5:45"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="seq_4.html#E7:45"><i><font color="Green" title="E37">A5</font></i></a>, <a class="txt" href="seq_4.html#E11:45"><i><font color="Green" title="E40">A8</font></i></a>, <a class="txt" href="seq_4.html#E12:45"><i><font color="Green" title="E41">A9</font></i></a>, <a class="ref" href="seq_2.html#T32" title="SEQ_2:th.32">SEQ_2:32</a></i>;<br/>


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