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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><a NAME="E1:48"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V3">bounded</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a><b> such that </b><br/><a NAME="E3:48"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> is <a href="seqm_3.html#V5">monotone</a>
 <b>by </b><i><a class="ref" href="seq_4.html#T56" target="_self">Th56</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>thus </b><a NAME="E4:48"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/>

<a NAME="E5:48"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i><a class="txt" href="seq_4.html#E1:48"><i><font color="Green">E39</font></i></a>, <a class="ref" href="seqm_3.html#T58">SEQM_3:58</a></i>;<br/>
<b>hence </b><a NAME="E6:48"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="seq_4.html#E3:48"><i><font color="Green">E40</font></i></a>, <a class="ref" href="seq_4.html#T53" target="_self">Th53</a></i>;<br/>


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