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

<b>let </b><font color="Maroon">f</font> be  <a href="asympt_0.html#V4" target="_self">eventually-positive</a> <a href="seq_1.html#NM1">Real_Sequence</a>, <font color="Maroon">g</font> be  <a href="asympt_0.html#V4" target="_self">eventually-positive</a> <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:38"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font> 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">f</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<b>set </b><font color="Maroon">l</font> =  <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> <font color="Maroon">g</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">delta</font> =  <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> <font color="Maroon">g</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c</font> = 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font></span>)</span></span>)</span>;<br/>
<a NAME="E2:38"/><i><font color="Green">E40</font></i>: 
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <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>consider </b><font color="Maroon">N</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="E4:38"/><i><font color="Green">E41</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="Maroon">N</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/> <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> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <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> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<a NAME="E5:38"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">f</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>consider </b><font color="Maroon">N1</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="E7:38"/><i><font color="Green">E43</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">N1</font> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">N2</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="E9:38"/><i><font color="Green">E55</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">N2</font> holds <br/><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">b</font> =  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N</font>,<font color="Maroon">N1</font>;<br/>
<b>set </b><font color="Maroon">a</font> =  <a href="asympt_0.html#K4" target="_self">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N</font>,<font color="Maroon">N1</font></span>)</span>,<font color="Maroon">N2</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_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="E1:38_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4" target="_self">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N</font>,<font color="Maroon">N1</font></span>)</span>,<font color="Maroon">N2</font>
 ;<br/><a NAME="E2:38_1"/><i><font color="Green">E69</font></i>: 
(  <a href="asympt_0.html#K4" target="_self">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N</font>,<font color="Maroon">N1</font></span>)</span>,<font color="Maroon">N2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N</font>,<font color="Maroon">N1</font> &amp;  <a href="asympt_0.html#K4" target="_self">max</a> <span class="p1">(<span class="default"><a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N</font>,<font color="Maroon">N1</font></span>)</span>,<font color="Maroon">N2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N2</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E3:38_1"/><i><font color="Green">E70</font></i>: 
(  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N</font>,<font color="Maroon">N1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font> &amp;  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N</font>,<font color="Maroon">N1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N1</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E4:38_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="asympt_0.html#K4" target="_self">max</a> <font color="Maroon">N</font>,<font color="Maroon">N1</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:38_1"/><b>then </b><i><font color="Green">E72</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N1</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N2</font> )
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:38_1"/><b>then </b>
 <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> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <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> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E7:38_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <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> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/><a NAME="E8:38_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font></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">1 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E9:38_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K18">"</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D9">SEQ_1:def 9</a></i>;<br/><a NAME="E10:38_1"/><b>then </b>
<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#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>;<br/><a NAME="E11:38_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<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#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D8">SEQ_1:def 8</a></i>;<br/><a NAME="E12:38_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">g</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>, </i>;<br/><a NAME="E13:38_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<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#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E14:38_1"/><b>then </b>
<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#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <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">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E15:38_1"/><b>then </b>
<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#K4">*</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><b>hence </b><a NAME="E16:38_1"/>
<font color="Maroon">f</font> <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">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><b>thus </b><a NAME="E17:38_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:38"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K9" target="_self">Big_Oh</a> <font color="Maroon">g</font>
 <b>by </b><i>, </i>;<br/>


</div>
