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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="E1:44"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="seqm_3.html#V5">monotone</a>
 <b>and </b><br/><a NAME="E2:44"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V3">bounded</a>
 ;<br/>

<a NAME="E3:44"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V2">bounded_below</a>
 
<b>by </b><i><a class="txt" href="seq_4.html#E2:44"><i><font color="Green">E38</font></i></a>, <a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a></i>;<br/>
<a NAME="E4:44"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V1">bounded_above</a>
 
<b>by </b><i><a class="txt" href="seq_4.html#E2:44"><i><font color="Green">E38</font></i></a>, <a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a></i>;<br/>
<a NAME="E5:44"/><b>then </b><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="seqm_3.html#V3">non-decreasing</a> implies <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V4">convergent</a> )
 
<b>by </b><i><a class="ref" href="seq_4.html#T51" target="_self">Th51</a></i>;<br/>
<a NAME="E6:44"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="seqm_3.html#V4">non-increasing</a> implies <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#E3:44"><i><font color="Green">E39</font></i></a>, <a class="ref" href="seq_4.html#T52" target="_self">Th52</a></i>;<br/>
<b>hence </b><a NAME="E7:44"/>
<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#E1:44"><i><font color="Green">E37</font></i></a>, <a class="txt" href="seq_4.html#E5:44"><i><font color="Green">E40</font></i></a>, <a class="ref" href="seqm_3.html#D7">SEQM_3:def 7</a></i>;<br/>


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