$\lim(a_n)\rightarrow 0$, $\lim(b_n)\rightarrow$ no limit, $a_nb_n\rightarrow 1$

Question

Find a sequence $a_n$ such that its $\lim(a_n)\rightarrow 0$ and $b_n$ that $\lim(b_n)$ has no limit (finite or infinite) such that $\lim(a_nb_n)\rightarrow1$
using arithmetic i need to find a sequence $a_n$ and $(a_n)^{-1}$.

let say $a_n$=$\frac{1}{n}$ what $b_n$ I need to choose?

Answer 1

Your sequence $a_n = n^{1/n}$ tends to $1$ instead of $0$ as you want.

Try the following sequence pair instead: $$a_n=1/n \text{ and } b_n=n$$

Another example:

$$a_n = (-1)^n/n \text{ and }b_n=(-1)^n n$$