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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:24"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V4" target="_self">convergent</a>
 <b>and </b><br/><a NAME="E2:24"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:24"/><i><font color="Green">E30</font></i>: 
not 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E4:24"/><i><font color="Green">E31</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="seq_2.html#E3:24"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:24"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="seq_2.html#E1:24"><i><font color="Green">E28</font></i></a>, <a class="ref" href="seq_2.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E7:24"/>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="seq_2.html#E6:24"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E8:24"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/>
<a NAME="E9:24"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>then </b> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>

<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="seq_2.html#E4:24"><i><font color="Green">E31</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><b>hence </b><a NAME="E3:24_1"/>
contradiction
 <b>by </b><i><a class="txt" href="seq_2.html#E6:24"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:24"/>
contradiction
 <b>by </b><i><a class="txt" href="seq_2.html#E2:24"><i><font color="Green">E29</font></i></a>, <a class="txt" href="seq_2.html#E9:24"><i><font color="Green">E33</font></i></a></i>;<br/>


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