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<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:66"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V3">bounded</a> &amp; ( <font color="Maroon">c<sub>3</sub></font> is <a href="limfunc1.html#V1" target="_self">divergent_to+infty</a> or <font color="Maroon">c<sub>3</sub></font> is <a href="limfunc1.html#V2" target="_self">divergent_to-infty</a> ) &amp; <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#NR1" target="_self">is_not_0</a>  )
 ;<br/>

<a NAME="E2:66"/><b>then </b><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K18">"</a>  is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="limfunc1.html#T61" target="_self">Th61</a></i>;<br/>
<a NAME="E3:66"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K11">(#)</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="seq_1.html#D9">SEQ_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:66"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="limfunc1.html#E1:66"><i><font color="Green">E39</font></i></a>, <a class="txt" href="limfunc1.html#E2:66"><i><font color="Green">E40</font></i></a>, <a class="ref" href="seq_2.html#T39">SEQ_2:39</a>, <a class="ref" href="seq_2.html#T40">SEQ_2:40</a></i>;<br/>


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