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

<b>reconsider </b><font color="Maroon">F</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span> as    <a href="fraenkel.html#M1">FUNCTION_DOMAIN</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="fraenkel.html#T10">FRAENKEL:10</a></i>;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E2:118"/>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1"><b>now </b></a><div class="add"><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/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:118_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</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="118_1_1_1"><b>suppose </b></a><a NAME="E1:118_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:118_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 &amp; <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">n</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:118_1_1_1"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <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">n</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_1_1_2"><b>suppose </b></a><a NAME="E1:118_1_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:118_1_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
( <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</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> &amp; <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">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 1 )
 <b>by </b><i/>;<br/><a NAME="E3:118_1_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E4:118_1_1_2"/>
<font color="Maroon">n</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#R1">&lt;=</a> <font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:118_1_1_2_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:118_1_1_2_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:118_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">n</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="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 1 <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="power.html#T31">POWER:31</a></i>;<br/><b>hence </b><a NAME="E3:118_1_1_2_1_1_1"/>
<font color="Maroon">n</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#R1">&lt;=</a> <font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 1
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:118_1_1_2_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:118_1_1_2_1_1_2"/>
<font color="Maroon">n</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#R1">&lt;=</a> <font color="Maroon">n</font> <a href="series_1.html#K3">to_power</a> 1
 <b>by </b><i><a class="ref" href="pre_ff.html#T10">PRE_FF:10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E5:118_1_1_2"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <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">n</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>set </b><font color="Maroon">t</font> =  <a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:118_2"/><i><font color="Green">E59</font></i>: 
 <a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="asympt_0.html#K18">to_power</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K9">Big_Oh</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span>
 ;<br/><b>consider </b><font color="Maroon">H</font> being    <a href="fraenkel.html#M1">FUNCTION_DOMAIN</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:118_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E4:118_2"/><i><font color="Green">E61</font></i>: 
(  <a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="asympt_0.html#K18">to_power</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K9">Big_Oh</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span> iff  ex <font color="Olive">N</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a> ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</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#NR2" target="_self">&gt;=</a> <font color="Olive">N</font> holds <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> &amp; <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">c</font> <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> <font color="Olive">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> ) ) ) )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">N0</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">c</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:118_2"/>
<font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E7:118_2"/><i><font color="Green">E62</font></i>: 
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#NR2" target="_self">&gt;=</a> <font color="Maroon">N0</font> holds <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> &amp; <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <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> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/><b>set </b><font color="Maroon">N</font> =  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,2;<br/><a NAME="E8:118_2"/><i><font color="Green">E63</font></i>: 
(  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N0</font> &amp;  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 <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="118_2_1"><b>now </b></a><div class="add"><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="E1:118_2_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4">max</a> <font color="Maroon">N0</font>,2
 ;<br/><a NAME="E2:118_2_1"/><b>then </b><i><font color="Green">E67</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N0</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:118_2_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i/>;<br/><a NAME="E4:118_2_1"/><i><font color="Green">E70</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="E5:118_2_1"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</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>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E6:118_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="power.html#T41">POWER:41</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:118_2"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:118"/>
( ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <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="Olive">n</font> ) &amp; not  <a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="asympt_0.html#K18">to_power</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K9">Big_Oh</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K4" target="_self">seq_const</a> 1</span>)</span></span>)</span> )
 <b>by </b><i/>;<br/>


</div>
