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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:2"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V4">convergent</a>
 <b>and </b><br/><a NAME="E2:2"/><i><font color="Green">E4</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:2"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>3</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>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="prob_2.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/>
<div><i><font color="Green">E6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:2_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:2_1"/><i><font color="Green">E7</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>6</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>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="prob_2.html#E4:2"><i><font color="Green">E5</font></i></a></i>;<br/><div><i><font color="Green">E8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:2_1_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<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>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<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>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>

<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>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prob_2.html#E2:2"><i><font color="Green">E4</font></i></a></i>
<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>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>

;<br/><b>hence </b><a NAME="E3:2_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="prob_2.html#E3:2_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="prob_2.html#E1:2_1_1"><i><font color="Green">E9</font></i></a></i>;<br/></div><b>end;</b></div><b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font>;<br/><b>thus </b><a NAME="E5:2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>7</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>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="prob_2.html#E4:2_1"><i><font color="Green">E8</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:2"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/>

<a NAME="E7:2"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="prob_2.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="prob_2.html#E4:2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E8:2"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="prob_2.html#E5:2"><i><font color="Green">E6</font></i></a>, <a class="txt" href="prob_2.html#E6:2"><i><font color="Green">E7</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>


</div>
