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<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> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:54"/><i><font color="Green">E42</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="E2:54"/><i><font color="Green">E43</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>1</sub></font> as   <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E4:54"/><i><font color="Green">E44</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>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E5:54"/><i><font color="Green">E45</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_4.html#T39">SEQ_4:39</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:54"/><i><font color="Green">E46</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>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="csspace4.html#E2:54"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E8:54"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>4</sub></font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="csspace4.html#E1:54"><i><font color="Green">E42</font></i></a>, <a class="ref" href="seq_4.html#T33">SEQ_4:33</a></i>;<br/>
<a NAME="E9:54"/><i><font color="Green">E48</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="csspace4.html#E1:54"><i><font color="Green">E42</font></i></a>, <a class="ref" href="seq_4.html#T33">SEQ_4:33</a></i>;<br/>
<div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:54_1"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>;<br/><a NAME="E2:54_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:54_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="csspace4.html#E7:54"><i><font color="Green">E46</font></i></a></i>;<br/><b>hence </b><a NAME="E4:54_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="csspace4.html#E4:54"><i><font color="Green">E44</font></i></a>, <a class="txt" href="csspace4.html#E1:54_1"><i><font color="Green">E50</font></i></a></i>;<br/></div><b>end;</b></div>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="seq_4.html#T41">SEQ_4:41</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="csspace4.html#E4:54"><i><font color="Green">E44</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E12:54"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="csspace4.html#E5:54"><i><font color="Green">E45</font></i></a>, <a class="txt" href="csspace4.html#E8:54"><i><font color="Green">E47</font></i></a>, <a class="txt" href="csspace4.html#E9:54"><i><font color="Green">E48</font></i></a>, <a class="txt" href="csspace4.html#E10:54"><i><font color="Green">E49</font></i></a>, <a class="ref" href="seq_2.html#T32">SEQ_2:32</a></i>;<br/>


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