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

<b>set </b><font color="Maroon">c<sub>1</sub></font> = 2 <a href="asympt_0.html#K7">(#)</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="asympt_1.html#K3" target="_self">seq_n^</a> 1;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="asympt_1.html#K1" target="_self">seq_a^</a> 2,2,0;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="asympt_1.html#K1" target="_self">seq_a^</a> 2,1,0 as  <a href="asympt_0.html#V4">eventually-positive</a> <a href="seq_1.html#NM1">Real_Sequence</a> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>thus </b><a NAME="E2:51"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="asympt_1.html#K1" target="_self">seq_a^</a> 2,1,0
 ;<br/>

<a NAME="E3:51"/><i><font color="Green">E34</font></i>: 
2 <a href="asympt_0.html#K7">(#)</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_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/><b>assume </b><a NAME="E1:51_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><b>thus </b><a NAME="E2:51_1"/>
<span class="p1">(<span class="default">2 <a href="asympt_0.html#K7">(#)</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span></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> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>;<br/><a NAME="E3:51_1"/><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</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>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1"><b>suppose </b></a><a NAME="E1:51_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:51_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</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>5</sub></font>
 <b>by </b><i><a class="ref" href="asympt_1.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_2"><b>suppose </b></a><a NAME="E1:51_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="series_1.html#K3">to_power</a> 1
<b>by </b><i><a class="ref" href="asympt_1.html#D3" target="_self">Def3</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E4:51_1"/>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="nat_1.html#K2">*</a> 0
 ;<br/><b>hence </b><a NAME="E5:51_1"/>
<span class="p1">(<span class="default">2 <a href="asympt_0.html#K7">(#)</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:51_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:51"/>
2 <a href="asympt_0.html#K7">(#)</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:51"><i><font color="Green">E34</font></i></a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_2"/>
 <a href="asympt_1.html#K1" target="_self">seq_a^</a> 2,2,0 <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:51_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="asympt_1.html#K1" target="_self">seq_a^</a> 2,2,0
 <b>and </b><br/><a NAME="E4:51_2"/><i><font color="Green">E36</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) ) )
 ;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:51_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E7:51_2"/><i><font color="Green">E38</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>7</sub></font> holds <br/>( <font color="Maroon">c<sub>5</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>6</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:51_2"><i><font color="Green">E36</font></i></a></i>;<br/><a NAME="E8:51_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">c<sub>8</sub></font> =  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K3">+</a> 1</span>)</span></span><a href="int_1.html#K2">\]</a></span>;<br/>

<a NAME="E1:51_2_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K3">+</a> 1</span>)</span></span><a href="int_1.html#K2">\]</a></span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E2:51_2_1"/>
<font color="Maroon">c<sub>8</sub></font> is   <a href="int_1.html#NM1">Integer</a>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T16">XXREAL_0:16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="asympt_1.html#E1:51_2_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<i><font color="Green">E40</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 0</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="asympt_1.html#D1" target="_self">Def1</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> 1
<b>by </b><i><a class="ref" href="power.html#T29">POWER:29</a></i>
;<br/>
<i><font color="Green">E41</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K1" target="_self">seq_a^</a> 2,2,0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="asympt_1.html#D1" target="_self">Def1</a></i>
<br/>.= 
2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>

;<br/>
<a NAME="E6:51_2_1"/><i><font color="Green">E42</font></i>: 
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="power.html#K3">to_power</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K3">+</a> 1</span>)</span></span><a href="int_1.html#K2">\]</a></span>
 
<b>by </b><i><a class="txt" href="asympt_1.html#E1:51_2_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="pre_ff.html#T10">PRE_FF:10</a></i>;<br/>
<a NAME="E7:51_2_1"/>
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K3">+</a> 1</span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</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="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/>
<a NAME="E8:51_2_1"/><b>then </b><i><font color="Green">E43</font></i>: 
2 <a href="power.html#K3">to_power</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K3">+</a> 1</span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="power.html#K4">to_power</a> <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#K3">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="pre_ff.html#T10">PRE_FF:10</a></i>;<br/>
2 <a href="power.html#K4">to_power</a> <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#K3">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</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="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E6:51_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="power.html#D3">POWER:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> 2
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
;<br/>
<a NAME="E10:51_2_1"/><b>then </b>
2 <a href="power.html#K4">to_power</a> <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#K3">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="asympt_1.html#E6:51_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="asympt_1.html#E4:51_2_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E11:51_2_1"/><b>then </b>
2 <a href="power.html#K3">to_power</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K3">+</a> 1</span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="asympt_1.html#E8:51_2_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:51_2_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K1" target="_self">seq_a^</a> 2,2,0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="asympt_1.html#E5:51_2_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="asympt_1.html#E6:51_2_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:51_2_1"/>
2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/>
<b>hence </b><a NAME="E14:51_2_1"/>
( <font color="Maroon">c<sub>8</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">c<sub>8</sub></font> is   <a href="nat_1.html#NM1">Nat</a> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:51_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="asympt_1.html#E12:51_2_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a>, <a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:51_2"/>
contradiction
 <b>by </b><i><a class="txt" href="asympt_1.html#E7:51_2"><i><font color="Green">E38</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:51"/>
not  <a href="asympt_1.html#K1" target="_self">seq_a^</a> 2,2,0 <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


</div>
