For any series that diverges, does there exist a sequence that converges to 0 yet the product diverges Suppose $\sum_{n=1}^{\infty} a_n = \infty$. Does there exists a sequence $\{b_n\}$ such that $ \lim_{n \rightarrow \infty} b_n = 0$ where $\sum_{n=1}^{\infty} a_n b_n = \infty$? I am able to prove this for the case when $a_n$ diverges to infinity, but other than that, I am not sure what to do and don't have a sense of whether this should be true or false. 

Hint: Take an increasing sequence $N(m)$ such that $\sum_{n=N(m-1)}^{k} a_n \ge 1$ for $k \ge N(m)$, and define $b_n$ in terms of the greatest $m$ such that $n \ge N(m)$.
