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

<b>assume </b><b>that </b><br/><a NAME="E1:46"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V2">bounded_below</a>
 <b>and </b><br/><a NAME="E2:46"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">seq</font> is <a href="seqm_3.html#V4">non-increasing</a>
 ;<br/>

<a NAME="E3:46"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, , </i>;<br/>
<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font>;<br/>
<b>consider </b><font color="Maroon">seq1</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E5:46"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<a NAME="E6:46"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">seq1</font> is <a href="funct_1.html#V3">constant</a>
 
<b>by </b><i><a class="txt" href="seq_4.html#E1"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="seqm_3.html#D6">SEQM_3:def 6</a></i>;<br/>
<a NAME="E7:46"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">seq1</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i/>;<br/>
<a NAME="E8:46"/>
<font color="Maroon">seq1</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="seq_4.html#E1"><i><font color="Green">Lemma40</font></i></a></i>;<br/>
<a NAME="E9:46"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="seq_4.html#E2"><i><font color="Green">Lemma41</font></i></a>, </i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <font color="Maroon">seq</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">m</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">seq2</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E11:46"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<a NAME="E12:46"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">seq2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="seq_4.html#T8" target="_self">Th8</a>, <a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>;<br/>
<a NAME="E13:46"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq2</font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:46"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">seq2</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, <a class="txt" href="seq_4.html#E1:46"><i><font color="Green">E45</font></i></a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:46_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">seq1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E1"><i><font color="Green">Lemma40</font></i></a></i>;<br/><a NAME="E2:46_1"/>
<font color="Maroon">seq2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_4.html#T8" target="_self">Th8</a></i>;<br/><b>hence </b><a NAME="E3:46_1"/>
<font color="Maroon">seq2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="txt" href="seq_4.html#E3:46"><i><font color="Green">E48</font></i></a>, <a class="ref" href="seqm_3.html#T13">SEQM_3:13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E16:46"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E3"><i><font color="Green">Lemma42</font></i></a>, <a class="txt" href="seq_4.html#E4"><i><font color="Green">Lemma43</font></i></a>, <a class="txt" href="seq_4.html#E2:46"><i><font color="Green">E46</font></i></a>, <a class="ref" href="seq_4.html#T9" target="_self">Th9</a>, <a class="ref" href="seq_2.html#T32">SEQ_2:32</a></i>;<br/>


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