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<b>let </b><font color="Maroon" title="c2">s</font>, <font color="Maroon" title="c3">s1</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:56"/><i><font color="Green" title="E31">A1</font></i>: 
( <font color="Maroon" title="c2">s</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> &amp; <font color="Maroon" title="c3">s1</font> is <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> &amp;  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  )
 ;<br/>

<a NAME="E2:56"/><b>then </b>
<font color="Maroon" title="c2">s</font> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <font color="Maroon" title="c3">s1</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>
 
<b>by </b><i><a class="ref" href="comseq_2.html#T42" target="_self" title="COMSEQ_2:th.42">Th42</a></i>;<br/>
<b>hence </b> <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <font color="Maroon" title="c3">s1</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> = 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <font color="Maroon" title="c3">s1</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_2.html#T11" target="_self" title="COMSEQ_2:th.11">Th11</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="comseq_2.html#E1:56"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="comseq_2.html#T43" target="_self" title="COMSEQ_2:th.43">Th43</a>, <a class="ref" href="complex1.html#T130" title="COMPLEX1:th.130">COMPLEX1:130</a></i>
;<br/><br/>


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