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 July 3rd, 2017, 07:25 AM #1 Senior Member   Joined: Dec 2015 From: Earth Posts: 232 Thanks: 26 Limit consequence Is consequence true ? If $\displaystyle \lim_{n\rightarrow \infty} \frac{a_n}{b_n}=0 \; \; \Rightarrow \; \; b_n\geq a_n$
 July 3rd, 2017, 08:17 AM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,327 Thanks: 2451 Math Focus: Mainly analysis and algebra No. $a_n = 0, b_n = -1$ If you meant to have both positive \begin{align*} a_n &= n \\ b_n &= \begin{cases} n^2 & (n \gt N) \\ 1 & (n \le N) \end{cases} \end{align*} The answer to the question you intended to ask (i.e. for sufficiently large $n$) is yes. Think about $\delta-\epsilon$ definition. $\displaystyle \lim_{n \to \infty} \frac{a_n}{b_n} = 0$ means that for every $\epsilon \gt 0$ there exists an $N \in \mathbb N$ such that $\frac{a_n}{b_n} \lt \epsilon$ for all $n \gt N$. Pick $\epsilon = 1$ and you have your result. Last edited by v8archie; July 3rd, 2017 at 08:40 AM.
 July 3rd, 2017, 08:39 AM #3 Senior Member   Joined: Dec 2015 From: Earth Posts: 232 Thanks: 26 i forgot to check $\displaystyle a_n,b_n >0$
 July 3rd, 2017, 08:40 AM #4 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,327 Thanks: 2451 Math Focus: Mainly analysis and algebra See update.
July 4th, 2017, 06:40 AM   #5
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Quote:
 Originally Posted by idontknow Is consequence true ? If $\displaystyle \lim_{n\rightarrow \infty} \frac{a_n}{b_n}=0 \; \; \Rightarrow \; \; b_n\geq a_n$
No.

$\displaystyle \left | \frac{a_{n}}{b_{n}} \right |\leq \epsilon\\ |a_{n}| \leq|b_n|$,
n>N

Ref: v8archie post#2.

Last edited by zylo; July 4th, 2017 at 06:51 AM. Reason: add n>N

 July 24th, 2017, 04:11 AM #6 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,240 Thanks: 884 Knowing the limit of a sequence does not tell you anything about all of the numbers in the sequence- you can change any finite number of terms in the sequence arbitrarily without changing the limit.

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