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<b>let </b><font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>let </b><font color="Maroon">seq</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:50"/><i><font color="Green">E53</font></i>: 
( ( <font color="Maroon">seq</font> is <a href="funct_1.html#V3">constant</a> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> ) or ( <font color="Maroon">seq</font> is <a href="funct_1.html#V3">constant</a> &amp;  ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> ) )
 ;<br/>

<div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>now </b></a><div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:50_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">seq</font> is <a href="funct_1.html#V3">constant</a>
 <b>and </b><br/><a NAME="E2:50_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font>
 ;<br/><b>consider </b><font color="Maroon">g1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> <b> such that </b><br/><a NAME="E4:50_1"/><i><font color="Green">E70</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">g1</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">g2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> <b> such that </b><br/><a NAME="E6:50_1"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:50_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E8:50_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">seq</font> is <a href="comseq_2.html#V2">convergent</a>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:50_1_1"/><i><font color="Green">E86</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 ;<br/><b>take </b><font color="Maroon">n</font> = 0;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E2:50_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E3:50_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><a NAME="E4:50_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">seq</font>
 <b>by </b><i><a class="ref" href="comseq_1.html#T2">COMSEQ_1:2</a></i>;<br/><a NAME="E5:50_1_1"/><b>then </b>
<font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:50_1_1"/><b>then </b>
<font color="Maroon">g2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font>
 <b>by </b><i/>;<br/><a NAME="E7:50_1_1"/><b>then </b>
<font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E8:50_1_1"/><b>then </b>
<font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:50_1_1"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <font color="Maroon">g</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="complex1.html#T130">COMPLEX1:130</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:50_1"/>
 <a href="comseq_2.html#K2">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="comseq_2.html#D5">COMSEQ_2:def 5</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><b>now </b></a><div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:50_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">seq</font> is <a href="funct_1.html#V3">constant</a>
 <b>and </b><br/><a NAME="E2:50_2"/><i><font color="Green">E88</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 ;<br/><b>consider </b><font color="Maroon">n</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="E4:50_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/><a NAME="E5:50_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><a NAME="E6:50_2"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">seq</font>
 <b>by </b><i><a class="ref" href="comseq_1.html#T2">COMSEQ_1:2</a></i>;<br/><b>hence </b><a NAME="E7:50_2"/>
 <a href="comseq_2.html#K2">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:50"/>
 <a href="comseq_2.html#K2">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i>, </i>;<br/>


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