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<b>let </b><font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c2">g</font> be  <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>assume </b><b>that </b><br/><a NAME="E1:58"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">g</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>and </b><br/><a NAME="E2:58"/><i><font color="Green" title="E30">A2</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1"/>
<font color="Maroon" title="c1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K12" title="ASYMPT_0:func.12">Big_Theta</a> <font color="Maroon" title="c2">g</font>
 ;<br/><a NAME="E2:58_1"/><b>then </b>
<font color="Maroon" title="c1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K11" title="ASYMPT_0:func.11">Big_Omega</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:58_1"/><b>then </b>
<font color="Maroon" title="c2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="ref" href="asympt_0.html#T19" target="_self" title="ASYMPT_0:th.19">Th19</a></i>;<br/><b>hence </b><a NAME="E4:58_1"/>
contradiction
 <b>by </b><i><a class="txt" href="asympt_0.html#E1:58"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="asympt_0.html#E2:58"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="asympt_0.html#T16" target="_self" title="ASYMPT_0:th.16">Th16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:58"/>
( <font color="Maroon" title="c1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K10" title="ASYMPT_0:func.10">Big_Oh</a> <font color="Maroon" title="c2">g</font> &amp; not <font color="Maroon" title="c1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="asympt_0.html#K12" title="ASYMPT_0:func.12">Big_Theta</a> <font color="Maroon" title="c2">g</font> )
 <b>by </b><i><a class="txt" href="asympt_0.html#E1:58"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="asympt_0.html#E2:58"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="asympt_0.html#T16" target="_self" title="ASYMPT_0:th.16">Th16</a></i>;<br/>


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