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<b>let </b><font color="Maroon" title="c2">seq1</font>, <font color="Maroon" title="c3">seq2</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:50"/><i><font color="Green" title="E33">A1</font></i>: 
( <font color="Maroon" title="c2">seq1</font> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a> &amp; <font color="Maroon" title="c3">seq2</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq2</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">n1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:50"/><i><font color="Green" title="E34">A2</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c4">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">seq2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">m</font>
 <b>by </b><i><a class="ref" href="limfunc1.html#T30" target="_self" title="LIMFUNC1:th.30">Th30</a></i>;<br/>
<a NAME="E4:50"/>
<font color="Maroon" title="c2">seq1</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">n1</font> is    <a href="seqm_3.html#M1" title="SEQM_3:mode.1">subsequence</a> of <font color="Maroon" title="c2">seq1</font>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T47" title="SEQM_3:th.47">SEQM_3:47</a></i>;<br/>
<a NAME="E5:50"/><b>then </b><i><font color="Green" title="E35">A3</font></i>: 
<font color="Maroon" title="c2">seq1</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">n1</font> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 
<b>by </b><i><a class="txt" href="limfunc1.html#E1:50"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="limfunc1.html#T53" target="_self" title="LIMFUNC1:th.53">Th53</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:50_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:50"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="xreal_1.html#T217" title="XREAL_1:th.217">XREAL_1:217</a></i>;<br/><b>let </b><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E2:50_1"/>
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">seq2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">n1</font></span>)</span>
 <b>by </b><i><a class="txt" href="limfunc1.html#E3:50"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><b>hence </b><a NAME="E3:50_1"/>
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:50"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">n1</font></span>)</span> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 
<b>by </b><i><a class="txt" href="limfunc1.html#E5:50"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="limfunc1.html#T49" target="_self" title="LIMFUNC1:th.49">Th49</a></i>;<br/>
<a NAME="E8:50"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">n1</font> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T42" title="SEQM_3:th.42">SEQM_3:42</a></i>;<br/>
<b>hence </b><a NAME="E9:50"/>
<font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq2</font> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 <b>by </b><i><a class="ref" href="limfunc1.html#T34" target="_self" title="LIMFUNC1:th.34">Th34</a></i>;<br/>


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