<?xml version="1.0"?>
<div class="add">

<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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> implies  ex <font color="Olive" title="b1">seq1</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> st <br/>( <font color="Olive" title="b1">seq1</font> is    <a href="seqm_3.html#M1" title="SEQM_3:mode.1">subsequence</a> of <font color="Maroon" title="c1">seq</font> &amp; <font color="Olive" title="b1">seq1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> ) )</span><br/>

<b>assume </b><a NAME="E1:48"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">seq1</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> st <br/>( <font color="Olive" title="b1">seq1</font> is    <a href="seqm_3.html#M1" title="SEQM_3:mode.1">subsequence</a> of <font color="Maroon" title="c1">seq</font> &amp; <font color="Olive" title="b1">seq1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> )</span><br/>

<b>consider </b><font color="Maroon" title="c2">Nseq</font> being  <a href="seqm_3.html#V1" title="SEQM_3:attr.1">increasing</a> <a href="seqm_3.html#NM1" title="SEQM_3:NM.1">Seq_of_Nat</a><b> such that </b><br/><a NAME="E3:48"/><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon" title="c1">seq</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">*</a> <font color="Maroon" title="c2">Nseq</font> is <a href="seqm_3.html#V5" title="SEQM_3:attr.5">monotone</a>
 <b>by </b><i><a class="ref" href="seq_4.html#T56" target="_self" title="SEQ_4:th.56">Th56</a></i>;<br/>
<b>take </b><font color="Maroon" title="c3">seq1</font> = <font color="Maroon" title="c1">seq</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">*</a> <font color="Maroon" title="c2">Nseq</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">seq1</font> is    <a href="seqm_3.html#M1" title="SEQM_3:mode.1">subsequence</a> of <font color="Maroon" title="c1">seq</font> &amp; <font color="Maroon" title="c3">seq1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> )</span><br/>

<b>thus </b><a NAME="E4:48"/>
<font color="Maroon" title="c3">seq1</font> is    <a href="seqm_3.html#M1" title="SEQM_3:mode.1">subsequence</a> of <font color="Maroon" title="c1">seq</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D10" title="SEQM_3:def.10">SEQM_3:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">seq1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a></span><br/>

<b>hence </b><a NAME="E5:48"/>
<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="txt" href="seq_4.html#E1:48"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="seq_4.html#E3:48"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="seq_4.html#T53" target="_self" title="SEQ_4:th.53">Th53</a>, <a class="ref" href="seqm_3.html#T58" title="SEQM_3:th.58">SEQM_3:58</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
