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<b>set </b><font color="Maroon">f</font> =  <a href="asympt_1.html#K1" target="_self">seq_a^</a> 2,1,0;<br/>
<b>reconsider </b><font color="Maroon">g</font> =  <a href="asympt_1.html#K1" target="_self">seq_a^</a> 2,1,1 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">g</font>
;<br/>

<b>thus </b><a NAME="E2:45"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="asympt_1.html#K1" target="_self">seq_a^</a> 2,1,1
 ;<br/>

<a NAME="E3:45"/><i><font color="Green">E49</font></i>: 
 <a href="asympt_0.html#K11">Big_Theta</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</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>  :  ex <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a> ex <font color="Olive">N</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( 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="Olive">N</font> holds <br/>( <font color="Olive">d</font> <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="Olive">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> &amp; <font color="Olive">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">c</font> <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="Olive">n</font></span>)</span> ) ) ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="asympt_0.html#T27">ASYMPT_0:27</a></i>;<br/>
<a NAME="E4:45"/><i><font color="Green">E50</font></i>: 
 <a href="asympt_1.html#K1" target="_self">seq_a^</a> 2,1,0 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/>
<a NAME="E5:45"/><i><font color="Green">E51</font></i>: 
2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_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="E1:45_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/><a NAME="E2:45_1"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="asympt_1.html#K1" target="_self">seq_a^</a> 2,1,0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 0</span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:45_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><b>then </b><span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</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> = 
2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="asympt_1.html#K1" target="_self">seq_a^</a> 2,1,0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E5:45_1"/>
<span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</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"><a href="asympt_1.html#K1" target="_self">seq_a^</a> 2,1,0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E6:45_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E7:45_1"/>
<span class="p1">(<span class="default"><a href="asympt_1.html#K1" target="_self">seq_a^</a> 2,1,0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <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="pre_ff.html#T10">PRE_FF:10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:45"/>
 <a href="asympt_1.html#K1" target="_self">seq_a^</a> 2,1,0 <a href="hidden.html#R2">in</a>  <a href="asympt_0.html#K11">Big_Theta</a> <font color="Maroon">g</font>
 <b>by </b><i>, , </i>;<br/>


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