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

<b>let </b><font color="Maroon">seq1</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">seq2</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:53"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">seq1</font> is <a href="limfunc1.html#V1" target="_self">divergent_to+infty</a> &amp; <font color="Maroon">seq2</font> is <a href="seq_2.html#V4">convergent</a> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq2</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">n1</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="E3:53"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">seq2</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:53"/>
<font color="Maroon">seq1</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n1</font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">seq1</font>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T47">SEQM_3:47</a></i>;<br/>
<a NAME="E5:53"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">seq1</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n1</font> is <a href="limfunc1.html#V1" target="_self">divergent_to+infty</a>
 
<b>by </b><i><a class="txt" href="limfunc1.html#E1"><i><font color="Green">Lemma32</font></i></a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:53_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">seq2</font></span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="txt" href="limfunc1.html#E1"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="xreal_1.html#T217">XREAL_1:217</a></i>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E2:53_1"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E3:53_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">seq2</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc1.html#E2"><i><font color="Green">Lemma33</font></i></a></i>;<br/><b>hence </b><a NAME="E4:53_1"/>
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">seq2</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:53"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n1</font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n1</font></span>)</span> is <a href="limfunc1.html#V1" target="_self">divergent_to+infty</a>
 
<b>by </b><i><a class="txt" href="limfunc1.html#E1:53"><i><font color="Green">E34</font></i></a>, </i>;<br/>
<a NAME="E8:53"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K11">(#)</a> <font color="Maroon">seq2</font></span>)</span> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n1</font> is <a href="limfunc1.html#V1" target="_self">divergent_to+infty</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T42">SEQM_3:42</a></i>;<br/>
<b>hence </b><a NAME="E9:53"/>
<font color="Maroon">seq1</font> <a href="seq_1.html#K11">(#)</a> <font color="Maroon">seq2</font> is <a href="limfunc1.html#V1" target="_self">divergent_to+infty</a>
 <b>by </b><i/>;<br/>


</div>
