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<b>let </b><font color="Maroon" title="c1">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:8"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="limfunc1.html#D4" title="LIMFUNC1:def.4">LIMFUNC1:def 4</a><br/>

<b>consider </b><font color="Maroon" title="c3">p</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="E3:8"/><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">r</font>
 <b>by </b><i><a class="ref" href="seq_4.html#T10" title="SEQ_4:th.10">SEQ_4:10</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="E5:8"/><i><font color="Green" title="E14">A3</font></i>: 
for <font color="Olive" title="b1">m</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="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">m</font>
 <b>by </b><i><a class="txt" href="heine.html#E1:8"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="limfunc1.html#D4" title="LIMFUNC1:def.4">LIMFUNC1:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">n</font>
;<br/>

<b>let </b><font color="Maroon" title="c5">m</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="E6:8"/>
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">n</font>
 ;<br/>

<a NAME="E7:8"/><b>then </b>
( <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">m</font> &amp; 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="heine.html#E5:8"><i><font color="Green" title="E14">A3</font></i></a></i>;<br/>
<a NAME="E8:8"/><b>then </b>
( 2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">m</font></span>)</span> &amp; 2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c3">p</font> &amp; 2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">p</font> )
 
<b>by </b><i><a class="ref" href="heine.html#T8" target="_self" title="HEINE:th.8">Th8</a>, <a class="ref" href="power.html#T44" title="POWER:th.44">POWER:44</a>, <a class="ref" href="power.html#T48" title="POWER:th.48">POWER:48</a></i>;<br/>
<a NAME="E9:8"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">m</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/>
<a NAME="E10:8"/><b>then </b>
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="heine.html#E3:8"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E11:8"/>
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="heine.html#K1" title="HEINE:func.1">to_power</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="ref" href="heine.html#D1" target="_self" title="HEINE:def.1">Def1</a></i>;<br/>


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