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, $a_{n}\to 0$.