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

<b>let </b><font color="Maroon">f</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:10_2_1"/>
<font color="Maroon">f</font> is <a href="asympt_0.html#V4" target="_self">eventually-positive</a>
 ;<br/>

<b>then 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="E3:10_2_1"/><i><font color="Green">E40</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">N</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#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:10_2_1"/>
<font color="Maroon">f</font> is <a href="asympt_0.html#V2" target="_self">eventually-nonnegative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">N</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="asympt_0.html#D4" target="_self">ASYMPT_0:def 4</a><br/>

<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="E1:10_2_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font>
 ;<br/>

<b>hence </b><a NAME="E2:10_2_1_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/>;<br/>


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

<b>thus </b><a NAME="E5:10_2_1"/>
<font color="Maroon">f</font> is <a href="asympt_0.html#V5" target="_self">eventually-nonzero</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">N</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="asympt_0.html#D7" target="_self">ASYMPT_0:def 7</a><br/>

<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="E1:10_2_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">N</font>
 ;<br/>

<b>hence </b><a NAME="E2:10_2_1_2"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i/>;<br/>


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


</div>
