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While answering Sequences that tend to zero, I came across the following related question:

Suppose we have a sequence $b_n$ whose limit is 0 as n tends to infinity.

Then can we always find another sequence $a_n$ such that $a_n$ also has 0 as the limit, but $\lim_{n->\infty}\frac{b_n}{a_n} = 0$?

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up vote 3 down vote accepted

Yes you can. Here's a simple example. Define a sequence $a_n$ such that $$a_n=\sqrt{|b_n|}$$. In that case, $a_n$ tends to $0$ and $\frac{b_n}{a_n}$ also tends to $0$.

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You have to watch out for $b_n=0$, but that is easy to fix. – Ross Millikan Jun 26 '14 at 15:38
Oh, I guess I was thinking in terms of series which is why this did not work. This is a correct answer for this question as I asked, but I wanted to ask something else. I will upvote this and change the question. – Wonder Jun 26 '14 at 15:39
Actually it is better to ask that separately I guess, I will accept this answer for here. – Wonder Jun 26 '14 at 15:39

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