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

<b>set </b><font color="Maroon" title="c1">f</font> =  <a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,0;<br/>
<b>set </b><font color="Maroon" title="c2">g</font> =  <a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,1;<br/>
<b>set </b><font color="Maroon" title="c3">h</font> = <span class="p1">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,0</span>)</span> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,1</span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">f</font> =  <a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,0 as  <a href="asympt_0.html#V4" title="ASYMPT_0:attr.4">eventually-positive</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c5">g</font> =  <a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,1 as  <a href="asympt_0.html#V4" title="ASYMPT_0:attr.4">eventually-positive</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> ;<br/>
<b>take </b>
<font color="Maroon" title="c4">f</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c5">g</font>
;<br/>

<b>thus </b><a NAME="E3:85"/>
( <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,0 &amp; <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,1 )
 ;<br/>

<a NAME="E4:85"/><i><font color="Green" title="E56">A1</font></i>: 
2 <a href="real_1.html#K2" title="REAL_1:func.2">"</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
;<br/>
<a NAME="E5:85"/><i><font color="Green" title="E57">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,0</span>)</span> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:85_1_1"/><i><font color="Green" title="E58">A3</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0</span>)</span>
 <b>by </b><i><a class="ref" href="asympt_1.html#D1" target="_self" title="ASYMPT_1:def.1">Def1</a></i>;<br/><a NAME="E2:85_1_1"/><i><font color="Green" title="E59">A4</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="asympt_1.html#D1" target="_self" title="ASYMPT_1:def.1">Def1</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,0</span>)</span> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font> = 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E1:85_1_1"><i><font color="Green" title="E58">A3</font></i></a>, <a class="txt" href="asympt_1.html#E10"><i><font color="Green" title="E7">Lm4</font></i></a></i>
<br/>.= 
2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E2:85_1_1"><i><font color="Green" title="E59">A4</font></i></a>, <a class="ref" href="power.html#T34" title="POWER:th.34">POWER:34</a></i>
<br/>.= 
2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">0 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span>

<br/>.= 
1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T33" title="POWER:th.33">POWER:33</a></i>
<br/>.= 
1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
<br/>.= 
2 <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 

;<br/><b>hence </b><a NAME="E4:85_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,0</span>)</span> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:85_1"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,0</span>)</span> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
 ;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E58">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="85_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">p</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:85_2"/><i><font color="Green" title="E59">A6</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><b>take </b><font color="Maroon" title="c7">N</font> = 0;<br/><b>let </b><font color="Maroon" title="c8">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E2:85_2"/>
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">N</font>
 ;<br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,0</span>)</span> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span> = 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E5:85"><i><font color="Green" title="E57">A2</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>
;<br/><b>hence </b><a NAME="E4:85_2"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,0</span>)</span> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <span class="p4">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:85_2"><i><font color="Green" title="E59">A6</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:85"/><b>then </b><i><font color="Green" title="E59">A7</font></i>: 
<span class="p1">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,0</span>)</span> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,1</span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#D6" title="SEQ_2:def.6">SEQ_2:def 6</a></i>;<br/>
<a NAME="E8:85"/><b>then </b>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,0</span>)</span> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K1" title="ASYMPT_1:func.1">seq_a^</a> 2,1,1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="asympt_1.html#E4:85"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="asympt_1.html#E6:85"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E9:85"/>
 <a href="asympt_0.html#K9" title="ASYMPT_0:func.9">Big_Oh</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_0.html#K9" title="ASYMPT_0:func.9">Big_Oh</a> <font color="Maroon" title="c5">g</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E7:85"><i><font color="Green" title="E59">A7</font></i></a>, <a class="ref" href="asympt_0.html#T15" title="ASYMPT_0:th.15">ASYMPT_0:15</a></i>;<br/>


</div>
