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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:45"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V1">bounded_above</a>
 <b>and </b><br/><a NAME="E2:45"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="seqm_3.html#V3">non-decreasing</a>
 ;<br/>

<a NAME="E3:45"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="seq_4.html#E1:45"><i><font color="Green">E37</font></i></a>, <a class="txt" href="seq_4.html#E2:45"><i><font color="Green">E38</font></i></a>, <a class="ref" href="seq_4.html#T51" target="_self">Th51</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as   <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E5:45"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E6:45"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E7:45"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="seq_4.html#E5:45"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E8:45"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="seq_4.html#T40" target="_self">Th40</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E10:45"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<a NAME="E11:45"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="seq_4.html#E10:45"><i><font color="Green">E43</font></i></a>, <a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>;<br/>
<a NAME="E12:45"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="seq_4.html#E3:45"><i><font color="Green">E39</font></i></a>, <a class="ref" href="seq_4.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E13:45"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="seq_4.html#E3:45"><i><font color="Green">E39</font></i></a>, <a class="txt" href="seq_4.html#E11:45"><i><font color="Green">E44</font></i></a>, <a class="ref" href="seq_4.html#T33" target="_self">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">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:45_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="seq_4.html#E5:45"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E2:45_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="seq_4.html#E10:45"><i><font color="Green">E43</font></i></a></i>;<br/><b>hence </b><a NAME="E3:45_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="seq_4.html#E2:45"><i><font color="Green">E38</font></i></a>, <a class="txt" href="seq_4.html#E1:45_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="seqm_3.html#T11">SEQM_3:11</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:45"/>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="seq_4.html#E6:45"><i><font color="Green">E41</font></i></a>, <a class="txt" href="seq_4.html#E8:45"><i><font color="Green">E42</font></i></a>, <a class="txt" href="seq_4.html#E12:45"><i><font color="Green">E45</font></i></a>, <a class="txt" href="seq_4.html#E13:45"><i><font color="Green">E46</font></i></a>, <a class="ref" href="seq_2.html#T32">SEQ_2:32</a></i>;<br/>


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