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

<span class="kw">let </span><font color="Maroon" title="c1">s</font>, <font color="Maroon" title="c2">s1</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">s</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp; <font color="Maroon" title="c2">s1</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp;  <a href="comseq_2.html#K3" title="COMSEQ_2:func.3">lim</a> <font color="Maroon" title="c1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  implies  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c2">s1</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:48"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
( <font color="Maroon" title="c1">s</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp; <font color="Maroon" title="c2">s1</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp;  <a href="comseq_2.html#K3" title="COMSEQ_2:func.3">lim</a> <font color="Maroon" title="c1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c2">s1</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/>

<a NAME="E2:48"/><span class="kw">then </span>
<font color="Maroon" title="c1">s</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c2">s1</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_2.html#T42" onmouseover="rs('comseq_2/T42')" onmouseout="rh()">COMSEQ_2:42</a></span>;<br/>
<span class="kw">hence </span> <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c2">s1</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> = 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_2.html#K3" title="COMSEQ_2:func.3">lim</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c2">s1</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#T27" target="_self" onmouseover="rs('seq_2/T27')" onmouseout="rh()">Th27</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="complex1.html#T44" onmouseover="rs('complex1/T44')" onmouseout="rh()">COMPLEX1:44</a>, <a class="ref" href="comseq_2.html#T43" onmouseover="rs('comseq_2/T43')" onmouseout="rh()">COMSEQ_2:43</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
