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<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="valued_0.html#M1" title="VALUED_0: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="E1:49"/><i><font color="Green" title="E31">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="valued_0.html#M1" title="VALUED_0: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="valued_0.html#V7" title="VALUED_0:attr.7">V30</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> <b> such that </b><br/><a NAME="E3:49"/><i><font color="Green" title="E32">A2</font></i>: 
<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>
 <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="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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">seq1</font> is    <a href="valued_0.html#M1" title="VALUED_0: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="E4:49"/>
<font color="Maroon" title="c3">seq1</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Maroon" title="c1">seq</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="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a></span><br/>

<b>thus </b><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>
 <b>by </b><i><a class="txt" href="#E1:49"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="#E3:49"><i><font color="Green" title="E32">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/>


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