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<b>let </b><font color="Maroon">a</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">h</font> be  <a href="fdiff_1.html#V1">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:7"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">a</font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">h</font>
 ;<br/>

<a NAME="E2:7"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">h</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">h</font> <a href="seq_1.html#NR1" target="_self">is_not_0</a>  )
 
<b>by </b><i><a class="ref" href="fdiff_1.html#D1">FDIFF_1:def 1</a></i>;<br/>
<a NAME="E3:7"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">a</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, <a class="ref" href="seq_4.html#T29">SEQ_4:29</a></i>;<br/>
<a NAME="E4:7"/><i><font color="Green">E46</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , <a class="ref" href="seq_4.html#T30">SEQ_4:30</a></i>;<br/>
<b>consider </b><font color="Maroon">I</font> being  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a><b> such that </b><br/><a NAME="E6:7"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">a</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">h</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:7_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:7_1"/>
<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/><b>hence </b><a NAME="E3:7_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:7"/><b>then </b>
<font color="Maroon">a</font> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 
<b>by </b><i><a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/>
<b>hence </b><a NAME="E9:7"/>
<font color="Maroon">a</font> is  <a href="fdiff_1.html#V1">convergent_to_0</a> <a href="seq_1.html#NM1">Real_Sequence</a>
 <b>by </b><i>, , <a class="ref" href="fdiff_1.html#D1">FDIFF_1:def 1</a></i>;<br/>


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