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Let $a_{n}$, $b_{n}$, be two sequences of positive real numbers such that $b_{n}< a_{n}$ for all $n\geq 1$, and $a_{n}\leq A$, for some $A>0$ and all $n$. I'm trying to define a new sequence $c_{n}$ in terms of $a_{n}$, $b_{n}$, or both such that

(1) $b_{n}c_{n}\to 0$, and

(2) $a_{n}c_{n}$ does not converges to $0$,

(3) $a_{n}c_{n}$ is bounded above (by something doesn't converge to 0)

I tried $c_{n}=\frac{(a_{n}+1)(b_{n}+2)}{(a_{n}+2)(b_{n}+1)}$, in this case (1) and (3) are true but (2) is not! Anyone has an idea?

Edit:I have a typo! I should write $\{b_{n}\} \subset \{a_{n}\}$, not $b_{n}< a_{n}$ Sorry!!

Also, $a_{n}\to 0$.

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    Andre's question is still valid.2012-06-10

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It is not hard to see that $c_n=\frac1{a_n}$ is the best you can do to have conditions (2) and (3) and the best possible chance for (1).

So, either $b_n/a_n$ converges to 0 and you have your sequence, or it doesn't and your sequence does not exist.

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    @Nick, that is precisely the point. You cannot expect to construct this $c_n$ unless you know that $b_n/a_n \to 0$.2012-06-10