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<b>set </b><font color="Maroon">c<sub>1</sub></font> =  <a href="asympt_1.html#K2" target="_self">seq_logn</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = <a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>;<br/>
<a NAME="E1:80"/>
( <a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="asympt_1.html#E23"><i><font color="Green">Lemma20</font></i></a></i>;<br/>
<b>then 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="E3:80"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">b<sub>1</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#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K19">/"</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="asympt_0.html#K4">max</a> 2,<font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E4:80"/><i><font color="Green">E55</font></i>: 
(  <a href="asympt_0.html#K4">max</a> 2,<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp;  <a href="asympt_0.html#K4">max</a> 2,<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:80_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4">max</a> 2,<font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:80_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:80"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:80_1"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K19">/"</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:80"><i><font color="Green">E54</font></i></a></i>;<br/><i><font color="Green">E58</font></i>: <span class="p1">(<span class="default"><a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E10"><i><font color="Green">Lemma7</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E4:80"><i><font color="Green">E55</font></i></a>, <a class="txt" href="asympt_1.html#E1:80_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="asympt_1.html#D2" target="_self">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E4:80"><i><font color="Green">E55</font></i></a>, <a class="txt" href="asympt_1.html#E1:80_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="asympt_1.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E75"><i><font color="Green">Lemma52</font></i></a></i>
;<br/><a NAME="E5:80_1"/><i><font color="Green">E59</font></i>: 
 <a href="square_1.html#K7">sqrt</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:80"><i><font color="Green">E55</font></i></a>, <a class="txt" href="asympt_1.html#E1:80_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="square_1.html#T93">SQUARE_1:93</a></i>;<br/><a NAME="E6:80_1"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:80_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E7:80_1"/><i><font color="Green">E61</font></i>: 
 <a href="square_1.html#K7">sqrt</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:80"><i><font color="Green">E55</font></i></a>, <a class="txt" href="asympt_1.html#E1:80_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="square_1.html#T93">SQUARE_1:93</a></i>;<br/><a NAME="E8:80_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:80_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="asympt_1.html#E5:80_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="asympt_1.html#E6:80_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E9:80_1"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E7:80_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E10:80_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E5:80_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E11:80_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 ;<br/><a NAME="E12:80_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/><a NAME="E13:80_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E14:80_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:80"/>
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>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">b<sub>2</sub></font> <a href="real_1.html#K6">/</a> 2
 ;<br/>


</div>
