Finding a sequence $a_{n}$ and $b_{n}$ where $a_{n}$ converges and $b_{n}$ is unbounded, and $a_{n}b_{n}$ converges

Question

So the question is to state without proof $a_{n}$ and $b_{n}$ such that $a_{n}$ converges, $b_{n}$ is unbounded and $a_{n}b_{n}$ converges.

I chose $a_{n} = \frac{1}{n}$ and $b_{n} = (-1)^n$

Then $a_{n}b_{n} = \frac{(-1)^n}{n}$

Am I correct?

Answer 1

From the comments, I now know that I made a silly error in choosing $b_{n} = (-1)^n$ which is bounded rather than unbounded, since $|b_{n}|=1$. Therefore instead choosing $a_{n}=\frac{1}{n^2}$ and $b_{n} = n$ would satisfy the original question, since $a_n b_n = \frac{1}{n}$ which converges.