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

<b>let </b><font color="Maroon" title="c1">f</font> be  <a href="asympt_0.html#V3" title="ASYMPT_0:attr.3">positive</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon" title="c2">t</font> be  <a href="asympt_0.html#V2" title="ASYMPT_0:attr.2">eventually-nonnegative</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:29_1"/>
<font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K9" title="ASYMPT_0:func.9">Big_Oh</a> <font color="Maroon" title="c1">f</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c3">s</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E3:29_1"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c2">t</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">s</font>
 <b>and </b><br/><a NAME="E4:29_1"/><i><font color="Green" title="E18">A2</font></i>: 
 ex <font color="Olive" title="b1">c</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ex <font color="Olive" title="b2">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>  st <br/>( <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; ( for <font color="Olive" title="b3">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>   st <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">N</font> holds <br/>( <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">n</font></span>)</span> &amp; <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 ) ) )
 ;<br/><b>consider </b><font color="Maroon" title="c4">c</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>, <font color="Maroon" title="c5">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> <b> such that </b><br/><a NAME="E6:29_1"/><i><font color="Green" title="E19">A3</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>and </b><br/><a NAME="E7:29_1"/><i><font color="Green" title="E20">A4</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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">N</font> holds <br/>( <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <font color="Maroon" title="c3">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="asympt_0.html#E4:29_1"><i><font color="Green" title="E18">A2</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1"/>
( <font color="Maroon" title="c5">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c5">N</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_1"/><i><font color="Green" title="E21">A5</font></i>: 
<font color="Maroon" title="c5">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c6">c</font> = <font color="Maroon" title="c4">c</font>;<br/><b>thus </b><a NAME="E2:29_1_1_1"/>
<font color="Maroon" title="c6">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="asympt_0.html#E6:29_1"><i><font color="Green" title="E19">A3</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c7">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>thus </b><a NAME="E3:29_1_1_1"/>
<font color="Maroon" title="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E3:29_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="asympt_0.html#E7:29_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="asympt_0.html#E1:29_1_1_1"><i><font color="Green" title="E21">A5</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_2"/><i><font color="Green" title="E21">A6</font></i>: 
<font color="Maroon" title="c5">N</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <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> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1</span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c6">B</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) where <font color="Olive" title="b1">n</font> is    <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>  : <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">N</font> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>from </b><i><a class="ref" href="asympt_0.html#S3" target="_self" title="ASYMPT_0:sch.3">ASYMPT_0:sch 3</a>(<a class="txt" href="asympt_0.html#E1:29_1_1_2"><i><font color="Green" title="E21">A6</font></i></a>);<br/></i><b>set </b><font color="Maroon" title="c7">b</font> =  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c6">B</font>;<br/><a NAME="E3:29_1_1_2"/><i><font color="Green" title="E22">A7</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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">N</font> holds <br/><font color="Maroon" title="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1"><b>proof </b></a><div class="add">

<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="E1:29_1_1_2_1"/>
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">N</font>
 ;<br/>

<a NAME="E2:29_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">t</font> <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#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 
;<br/>
<a NAME="E3:29_1_1_2_1"/><b>then </b><i><font color="Green" title="E23">A8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">t</font> <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#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/>
<a NAME="E4:29_1_1_2_1"/><i><font color="Green" title="E24">A9</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="asympt_0.html#D5" target="_self" title="ASYMPT_0:def.5">Def5</a></i>;<br/>
<a NAME="E5:29_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">t</font> <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#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="asympt_0.html#E3:29_1_1_2_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<b>hence </b><a NAME="E6:29_1_1_2_1"/>
<font color="Maroon" title="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E4:29_1_1_2_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E4:29_1_1_2"/>
 ex <font color="Olive" title="b1">c</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; ( for <font color="Olive" title="b2">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  <font color="Maroon" title="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">n</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_2_2_1"/>
(  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c6">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">c</font> or  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c6">B</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">c</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_2_2_1_1"/><i><font color="Green" title="E23">A10</font></i>: 
 <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c6">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">c</font>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon" title="c4">c</font>
;<br/><b>thus </b><a NAME="E2:29_1_1_2_2_1_1"/>
<font color="Maroon" title="c4">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="asympt_0.html#E6:29_1"><i><font color="Green" title="E19">A3</font></i></a></i>;<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>thus </b><a NAME="E3:29_1_1_2_2_1_1"/>
<font color="Maroon" title="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_2_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_2_2_1_1_1_1"/>
( <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">N</font> or <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">N</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_2_2_1_1_1_1_1"/><i><font color="Green" title="E24">A11</font></i>: 
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">N</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_2_2_1_1_1_1_1"/>
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="asympt_0.html#D5" target="_self" title="ASYMPT_0:def.5">Def5</a></i>;<br/><a NAME="E3:29_1_1_2_2_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="asympt_0.html#E3:29_1_1_2"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="asympt_0.html#E1:29_1_1_2_2_1_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="txt" href="asympt_0.html#E1:29_1_1_2_2_1_1_1_1_1"><i><font color="Green" title="E24">A11</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_2_2_1_1_1_1_1"/>
<font color="Maroon" title="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_2_2_1_1_1_1_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="c5">N</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:29_1_1_2_2_1_1_1_1_2"/>
<font color="Maroon" title="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E3:29_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="asympt_0.html#E7:29_1"><i><font color="Green" title="E20">A4</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_2_2_1_2"/><i><font color="Green" title="E23">A12</font></i>: 
 <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c6">B</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">c</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c8">b</font> =  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c6">B</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><b>take </b>
<font color="Maroon" title="c8">b</font>
;<br/><b>thus </b><a NAME="E3:29_1_1_2_2_1_2"/>
<font color="Maroon" title="c8">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="asympt_0.html#E6:29_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="asympt_0.html#E1:29_1_1_2_2_1_2"><i><font color="Green" title="E23">A12</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c9">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>thus </b><a NAME="E4:29_1_1_2_2_1_2"/>
<font color="Maroon" title="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_2_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_2_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="E1:29_1_1_2_2_1_2_1_1"/>
( <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">N</font> or <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">N</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_2_2_1_2_1_1_1"/>
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">N</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:29_1_1_2_2_1_2_1_1_1"/>
<font color="Maroon" title="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="asympt_0.html#E3:29_1_1_2"><i><font color="Green" title="E22">A7</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E24">A13</font></i>: 
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">N</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_2_2_1_2_1_1_2"/>
<font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="asympt_0.html#D5" target="_self" title="ASYMPT_0:def.5">Def5</a></i>;<br/><a NAME="E3:29_1_1_2_2_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; <font color="Maroon" title="c4">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="asympt_0.html#E3:29_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="asympt_0.html#E7:29_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="asympt_0.html#E1:29_1_1_2_2_1_2"><i><font color="Green" title="E23">A12</font></i></a>, <a class="txt" href="asympt_0.html#E1:29_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E24">A13</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_2_2_1_2_1_1_2"/>
<font color="Maroon" title="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>given </b><font color="Maroon" title="c3">c</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><a NAME="E2:29"/><i><font color="Green" title="E17">A14</font></i>: 
( <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; ( 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  <font color="Maroon" title="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) )
 ;<br/>

<a NAME="E3:29"/><i><font color="Green" title="E18">A15</font></i>: 
<font color="Maroon" title="c2">t</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">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> <b> such that </b><br/><a NAME="E5:29"/><i><font color="Green" title="E19">A16</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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">N</font> holds <br/><font color="Maroon" title="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="asympt_0.html#D4" target="_self" title="ASYMPT_0:def.4">Def4</a></i>;<br/>
<a NAME="E6:29"/>
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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">N</font> holds <br/>( <font color="Maroon" title="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <font color="Maroon" title="c2">t</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 
<b>by </b><i><a class="txt" href="asympt_0.html#E2:29"><i><font color="Green" title="E17">A14</font></i></a>, <a class="txt" href="asympt_0.html#E5:29"><i><font color="Green" title="E19">A16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:29"/>
<font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K9" title="ASYMPT_0:func.9">Big_Oh</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E2:29"><i><font color="Green" title="E17">A14</font></i></a>, <a class="txt" href="asympt_0.html#E3:29"><i><font color="Green" title="E18">A15</font></i></a></i>;<br/>


</div>
