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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:61"/><i><font color="Green">E39</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:61"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; ( 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#NR4" target="_self">&gt;</a> 0 holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 ;<br/>

<a NAME="E3:61"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="asympt_0.html#V4">eventually-positive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_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">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:61_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/>

<a NAME="E2:61_1"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>3</sub></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="E3:61_1"/><b>then </b>
 <a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>3</sub></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="E4:61_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K2">*</a> 0
 
<b>by </b><i><a class="txt" href="asympt_1.html#E1:61_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<b>hence </b><a NAME="E5:61_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:61"><i><font color="Green">E40</font></i></a>, <a class="txt" href="asympt_1.html#E1:61_1"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font>
;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></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> <span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="asympt_1.html#K2" target="_self">seq_logn</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E5:61"/><i><font color="Green">E41</font></i>: 
( <a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="asympt_1.html#E1:61"><i><font color="Green">E39</font></i></a>, <a class="txt" href="asympt_1.html#E23"><i><font color="Green">Lemma20</font></i></a></i>;<br/>
<a NAME="E6:61"/>
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> 1 holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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> <span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:61_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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> <span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</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#K3" target="_self">seq_n^</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</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"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></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> <span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E2:61"><i><font color="Green">E40</font></i></a>, <a class="txt" href="asympt_1.html#E1:61_2"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E1:61_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="asympt_1.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="series_1.html#K3">to_power</a> 1</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">c<sub>9</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="series_1.html#K3">to_power</a> 1</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">c<sub>9</sub></font></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">c<sub>9</sub></font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></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"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></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"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E1:61_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="power.html#T34">POWER:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></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">c<sub>9</sub></font> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E1:61_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="power.html#T33">POWER:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E1:61_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="asympt_1.html#D2" target="_self">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</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#K3" target="_self">seq_n^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E1:61_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="asympt_1.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="asympt_1.html#E10"><i><font color="Green">Lemma7</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E3:61_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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> <span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="asympt_1.html#K2" target="_self">seq_logn</a>  <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><a href="asympt_1.html#K3" target="_self">seq_n^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:61"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></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> <span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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> <span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="asympt_1.html#E5:61"><i><font color="Green">E41</font></i></a>, <a class="txt" href="asympt_1.html#E53"><i><font color="Green">Lemma35</font></i></a></i>;<br/>
<a NAME="E8:61"/><b>then </b>
( <font color="Maroon">c<sub>3</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#K3" target="_self">seq_n^</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; not  <a href="asympt_1.html#K3" target="_self">seq_n^</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="asympt_0.html#T16">ASYMPT_0:16</a></i>;<br/>
<a NAME="E9:61"/><b>then </b>
( <font color="Maroon">c<sub>3</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#K3" target="_self">seq_n^</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; not <font color="Maroon">c<sub>3</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#K3" target="_self">seq_n^</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></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="E10:61"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">c<sub>3</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#K3" target="_self">seq_n^</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; not  <a href="asympt_0.html#K9">Big_Oh</a> <font color="Maroon">c<sub>3</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#K3" target="_self">seq_n^</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="asympt_1.html#T4" target="_self">Th4</a></i>;<br/>


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