real analysis - If a ${a_n}$ diverges, so $a_n rightarrow + infty$, how to find sequence ${b_n}$ such that $sum |b_n|

If we are given any sequence of real numbers $\{a_n\}$ diverges, so $a_n \rightarrow + \infty$, how can we find a sequence $\{b_n\}$ such that $\sum |b_n|$ converges but $\sum |a_n||b_n|$ diverges?

I want to use this fact in another problem but don't immediately see how to prove it.

Answer

Just pick a subsequence such that $a_{n_k} > k$, and then let $b_{n_k} = 1/k^2$, and $b_n = 0$ if $n$ is not an index in that subsequence.