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

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="valued_0.html#M2" title="VALUED_0:mode.2">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/>

<span class="kw">assume </span><a NAME="E1:49"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="valued_0.html#M2" title="VALUED_0:mode.2">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/>

<span class="kw">consider </span><font color="Maroon" title="c2">Nseq</font> being  <a href="valued_0.html#V7" title="VALUED_0:attr.7">V35</a>() <a href="nat_1.html#NM2" title="NAT_1:NM.2">sequence</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a><span class="kw"> such that </span><br/><a NAME="E3:49"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c1">seq</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c2">Nseq</font> is  <a href="seqm_3.html#V1" title="SEQM_3:attr.1">monotone</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T39" target="_self" title="SEQ_4:th.39">Th56</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c3">seq1</font> = <font color="Maroon" title="c1">seq</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c2">Nseq</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">seq1</font> is    <a href="valued_0.html#M2" title="VALUED_0:mode.2">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/>

<span class="kw">thus </span><a NAME="E4:49"/>
<font color="Maroon" title="c3">seq1</font> is    <a href="valued_0.html#M2" title="VALUED_0:mode.2">subsequence</a> of <font color="Maroon" title="c1">seq</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">thus </span><a NAME="E5:49"/>
<font color="Maroon" title="c3">seq1</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E3:49"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="seq_4.html#T36" target="_self" title="SEQ_4:th.36">Th53</a>, <a class="ref" href="seqm_3.html#T29" title="SEQM_3:th.29">SEQM_3:29</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
