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

<b>set </b><font color="Maroon">c<sub>1</sub></font> =  <a href="asympt_1.html#K5" target="_self">seq_n!</a> 0;<br/>
<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> =  <a href="asympt_1.html#K1" target="_self">seq_a^</a> 2,2,0 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>2</sub></font>
;<br/>

<b>thus </b><a NAME="E2:87"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="asympt_1.html#K1" target="_self">seq_a^</a> 2,2,0
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K5" target="_self">seq_n!</a> 0</span>)</span>;<br/>
<div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</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="E1:87_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E2:87_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K5">"</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/><a NAME="E3:87_1"/><b>then </b><i><font color="Green">E57</font></i>: 
1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="xxreal_0.html#K4">max</a> 10,<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">9 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="power.html#K5">log</a> 2,<span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span>;<br/><a NAME="E4:87_1"/><i><font color="Green">E58</font></i>: 
(  <a href="xxreal_0.html#K4">max</a> 10,<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">9 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="power.html#K5">log</a> 2,<span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 10 &amp;  <a href="xxreal_0.html#K4">max</a> 10,<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">9 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="power.html#K5">log</a> 2,<span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> <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">9 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="power.html#K5">log</a> 2,<span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></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="E5:87_1"/>
 <a href="xxreal_0.html#K4">max</a> 10,<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">9 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="power.html#K5">log</a> 2,<span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> 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>6</sub></font> =  <a href="xxreal_0.html#K4">max</a> 10,<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">9 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="power.html#K5">log</a> 2,<span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="asympt_1.html#E4:87_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E7:87_1"/>
<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>
 ;<br/><a NAME="E8:87_1"/><b>then </b><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 10 &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">9 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="power.html#K5">log</a> 2,<span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> )
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:87_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><i><font color="Green">E60</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K5" target="_self">seq_n!</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</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#K5" target="_self">seq_n!</a> 0</span>)</span> <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#E10"><i><font color="Green">Lemma7</font></i></a></i>
<br/>.= 
<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>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 0</span>)</span></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#K5" target="_self">seq_n!</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="asympt_1.html#D1" target="_self">Def1</a></i>
<br/>.= 
<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>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 0</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 0</span>)</span> <a href="newton.html#K12">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="asympt_1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="newton.html#K12">!</a> </span>)</span>

;<br/><a NAME="E10:87_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K5" target="_self">seq_n!</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</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>8</sub></font> <a href="real_1.html#K5">-</a> 9</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E8:87_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="asympt_1.html#E70"><i><font color="Green">Lemma47</font></i></a></i>;<br/><a NAME="E11:87_1"/>
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">9 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="power.html#K5">log</a> 2,<span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 9 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> 2,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/><a NAME="E12:87_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 9 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> 2,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E8:87_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:87_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> 9 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="power.html#K5">log</a> 2,<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E14:87_1"/><b>then </b>
2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> 9</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> 2,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_ff.html#T10">PRE_FF:10</a></i>;<br/><a NAME="E15:87_1"/><b>then </b>
2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> 9</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:87_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="power.html#D3">POWER:def 3</a></i>;<br/><a NAME="E16:87_1"/><b>then </b>
1 <a href="real_1.html#K6">/</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>8</sub></font> <a href="real_1.html#K5">-</a> 9</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_1.html#E3:87_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="real_2.html#T152">REAL_2:152</a></i>;<br/><a NAME="E17:87_1"/><b>then </b>
1 <a href="real_1.html#K6">/</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>8</sub></font> <a href="real_1.html#K5">-</a> 9</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T56">XCMPLX_1:56</a></i>;<br/><a NAME="E18:87_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K5" target="_self">seq_n!</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E10:87_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E19:87_1"/>
 <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"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K19">/"</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K5" target="_self">seq_n!</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a>, <a class="txt" href="asympt_1.html#E9:87_1"><i><font color="Green">E60</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:87"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K5" target="_self">seq_n!</a> 0</span>)</span> 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="E5:87"/><b>then </b>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K5" target="_self">seq_n!</a> 0</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="asympt_1.html#E3:87"><i><font color="Green">E56</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<a NAME="E6:87"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <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#K5" target="_self">seq_n!</a> 0</span>)</span> &amp; not  <a href="asympt_1.html#K5" target="_self">seq_n!</a> 0 <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="asympt_1.html#E4:87"><i><font color="Green">E57</font></i></a>, <a class="ref" href="asympt_0.html#T16">ASYMPT_0:16</a></i>;<br/>
<a NAME="E7:87"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <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#K5" target="_self">seq_n!</a> 0</span>)</span> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K10">Big_Omega</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K5" target="_self">seq_n!</a> 0</span>)</span> )
 
<b>by </b><i><a class="ref" href="asympt_0.html#T19">ASYMPT_0:19</a></i>;<br/>
<b>hence </b><a NAME="E8:87"/>
(  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="asympt_0.html#K9">Big_Oh</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K5" target="_self">seq_n!</a> 0</span>)</span> &amp; not  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="asympt_0.html#K9">Big_Oh</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K5" target="_self">seq_n!</a> 0</span>)</span> )
 <b>by </b><i><a class="ref" href="asympt_1.html#T4" target="_self">Th4</a></i>;<br/>


</div>
