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<b>let </b><font color="Maroon">seq</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/><b>let </b><font color="Maroon">seq1</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:46"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">seq</font> is <a href="comseq_2.html#V2">convergent</a>
 <b>and </b><br/><a NAME="E2:46"/><i><font color="Green">E55</font></i>: 
 ex <font color="Olive">k</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="funct_2.html#R2">=</a> <font color="Maroon">seq1</font> <a href="comseq_3.html#K6">^\</a> <font color="Olive">k</font>
 ;<br/>

<a NAME="E3:46"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">seq1</font> is <a href="comseq_2.html#V2">convergent</a>
 
<b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">k</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="E5:46"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">seq</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">seq1</font> <a href="comseq_3.html#K6">^\</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_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="E1:46_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 ;<br/><b>then consider </b><font color="Maroon">n1</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="E3:46_1"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><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="Olive">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">seq</font></span>)</span></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="comseq_2.html#D5">COMSEQ_2:def 5</a></i>;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>;<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="E4:46_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><b>then consider </b><font color="Maroon">l</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:46_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">l</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E8:46_1"/>
<font color="Maroon">m</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">m1</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="E10:46_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font>
 ;<br/><div><i><font color="Green">E86</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_1_1"/>
not <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font>
 ;<br/><a NAME="E2:46_1_1"/><b>then </b>
<font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E3:46_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E12:46_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/><a NAME="E13:46_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">m1</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">seq</font></span>)</span></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>, </i>;<br/><b>hence </b><a NAME="E14:46_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">seq1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">seq</font></span>)</span></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="comseq_3.html#D6">COMSEQ_3:def 6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:46"/>
 <a href="comseq_2.html#K2">lim</a> <font color="Maroon">seq1</font> <a href="hidden.html#R1">=</a>  <a href="comseq_2.html#K2">lim</a> <font color="Maroon">seq</font>
 <b>by </b><i>, <a class="ref" href="comseq_2.html#D5">COMSEQ_2:def 5</a></i>;<br/>


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