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<b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:31"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a>
 <b>and </b><br/><a NAME="E2:31"/><i><font color="Green">E46</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:31"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 <b>by </b><i>, , </i>;<br/>
<b>take </b>
<font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font>
;<br/>

<b>thus </b><a NAME="E5:31"/>
( <font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">seq</font> &amp; <font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font> <a href="seq_1.html#NR1" target="_self">is_not_0</a>  )
 <b>by </b><i>, <a class="ref" href="seqm_3.html#T47">SEQM_3:47</a></i>;<br/>


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