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

<b>let </b><font color="Maroon">f</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:63"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="series_1.html#K3">to_power</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Olive">n</font></span>)</span> ) )
 ;<br/>

<b>set </b><font color="Maroon">g</font> =  <a href="asympt_1.html#K3" target="_self">seq_n^</a> 8;<br/>
<b>set </b><font color="Maroon">h</font> = <font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 8</span>)</span>;<br/>
<a NAME="E2:63"/>
<font color="Maroon">f</font> is <a href="asympt_0.html#V4">eventually-positive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>take </b>
2
; <i><font color="Red">:: according to </font></i><a class="ref" href="asympt_0.html#D6">ASYMPT_0:def 6</a><br/>

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:63_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/>

<a NAME="E2:63_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>then </b><i><font color="Green">E51</font></i>: <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
;<br/>
<a NAME="E4:63_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 2 <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/>
<a NAME="E5:63_1"/>
 <a href="power.html#K6">log</a> 2,<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="power.html#K6">log</a> 2,2
 
<b>by </b><i>, <a class="ref" href="pre_ff.html#T12">PRE_FF:12</a></i>;<br/>
<a NAME="E6:63_1"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="power.html#T60">POWER:60</a></i>;<br/>
<a NAME="E7:63_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</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="E8:63_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 2</span>)</span> <a href="nat_1.html#K2">*</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<b>hence </b><a NAME="E9:63_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> 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">f</font>
;<br/>

<div><i><font color="Green">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</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="E1:63_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>set </b><font color="Maroon">N</font> =  <a href="xxreal_0.html#K4">max</a> 3,<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 6</span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span>;<br/><a NAME="E3:63_2"/><i><font color="Green">E61</font></i>: 
(  <a href="xxreal_0.html#K4">max</a> 3,<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 6</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> 3 &amp;  <a href="xxreal_0.html#K4">max</a> 3,<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 6</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"><font color="Maroon">p</font> <a href="power.html#K3">to_power</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 6</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="E4:63_2"/>
 <a href="xxreal_0.html#K4">max</a> 3,<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 6</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">N</font> =  <a href="xxreal_0.html#K4">max</a> 3,<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 6</span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><b>take </b><font color="Maroon">N</font> = <font color="Maroon">N</font>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E6:63_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font>
 ;<br/><a NAME="E7:63_2"/><b>then </b><i><font color="Green">E63</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3 &amp; <font color="Maroon">n</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"><font color="Maroon">p</font> <a href="power.html#K3">to_power</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 6</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#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:63_2"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="power.html#K6">log</a> 2,<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="power.html#K6">log</a> 2,3
 <b>by </b><i><a class="ref" href="pre_ff.html#T12">PRE_FF:12</a></i>;<br/><a NAME="E9:63_2"/>
 <a href="power.html#K6">log</a> 2,3 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="power.html#K6">log</a> 2,2
 <b>by </b><i><a class="ref" href="power.html#T65">POWER:65</a></i>;<br/><a NAME="E10:63_2"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="power.html#K6">log</a> 2,2
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E11:63_2"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="power.html#K6">log</a> 2,<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="ref" href="power.html#T60">POWER:60</a></i>;<br/><a NAME="E12:63_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:63_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 6 <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/><i><font color="Green">E71</font></i>: <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 8</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</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> 8</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></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#K3" target="_self">seq_n^</a> 8</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="txt" href="asympt_1.html#E1"><i><font color="Green">Lemma47</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 8</span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 8</span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1">(<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">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 2</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 8</span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 8</span>)</span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 8</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K5">-</a> 8</span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="power.html#T34">POWER:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 6</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 6</span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="power.html#T33">POWER:33</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 6</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 6</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">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T103">XCMPLX_1:103</a></i>
;<br/><a NAME="E15:63_2"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 6</span>)</span> <a href="nat_1.html#K2">*</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 6</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">n</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E16:63_2"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 8</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 6</span>)</span>
 <b>by </b><i>, <a class="txt" href="asympt_1.html#E1:63"><i><font color="Green">E49</font></i></a>, <a class="ref" href="real_2.html#T151">REAL_2:151</a></i>;<br/><a NAME="E17:63_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">p1</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 6</span>)</span></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/><a NAME="E18:63_2"/>
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="power.html#K3">to_power</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 6</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> <font color="Maroon">p</font> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 6</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="E19:63_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p</font> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 6</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E20:63_2"/><b>then </b>
<font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 6 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 6</span>)</span></span>)</span></span>)</span> <a href="power.html#K3">to_power</a> 6
 <b>by </b><i>, </i>;<br/><a NAME="E21:63_2"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 6 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p1</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 6</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 6</span>)</span>
 <b>by </b><i>, <a class="ref" href="power.html#T38">POWER:38</a></i>;<br/><a NAME="E22:63_2"/>
<font color="Maroon">p1</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</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/><a NAME="E23:63_2"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 6</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_2.html#T152">REAL_2:152</a></i>;<br/><a NAME="E24:63_2"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 6</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="power.html#K4">to_power</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="power.html#T33">POWER:33</a></i>;<br/><a NAME="E25:63_2"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 6</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p1</font></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/><a NAME="E26:63_2"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 6</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T56">XCMPLX_1:56</a></i>;<br/><a NAME="E27:63_2"/><b>then </b><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 8</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E28:63_2"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 6</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">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 6</span>)</span> <a href="nat_1.html#K2">*</a> 0
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E29:63_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 6</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">n</font></span>)</span></span>)</span> <a href="real_1.html#K2">"</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="E30:63_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 8</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:63"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>;<br/><b>hence </b><a NAME="E31:63_2"/>
 <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">f</font> <a href="seq_1.html#K19">/"</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 8</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:63"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 8</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="E6:63"/><b>then </b>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 8</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<a NAME="E7:63"/><b>then </b>
( <font color="Maroon">f</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#K3" target="_self">seq_n^</a> 8</span>)</span> &amp; not  <a href="asympt_1.html#K3" target="_self">seq_n^</a> 8 <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="ref" href="asympt_0.html#T16">ASYMPT_0:16</a></i>;<br/>
<a NAME="E8:63"/><b>then </b>
( <font color="Maroon">f</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#K3" target="_self">seq_n^</a> 8</span>)</span> &amp; not <font color="Maroon">f</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#K3" target="_self">seq_n^</a> 8</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="E9:63"/>
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp;  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">f</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#K3" target="_self">seq_n^</a> 8</span>)</span> &amp; not  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">f</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#K3" target="_self">seq_n^</a> 8</span>)</span> )
 <b>by </b><i/>;<br/>


</div>
