Suppose $\sum_{i = 1} a_i \sum_{i = 1} b_i < \infty$. If $\sum_{i = 1} a_i = \infty$, what does it say about $(b_i)$?
Yes, it does. If the sum of the $b_i$'s is positive, then its product with the sum of the $a_i$'s is also infinite, contradicting the assumption.