Suppose $\sum_{i = 1} a_i \sum_{i = 1} b_i < \infty$. If $\sum_{i = 1} a_i = \infty$, what does it say about $(b_i)$?

Question

Let $(a_i), (b_i)$ be two non-negative sequence.

Suppose $\sum_{i = 1} a_i \sum_{i = 1} b_i < \infty$. If $\sum_{i = 1} a_i = \infty$, what does it say about $(b_i)$?

Does it necessarily mean that $\sum_{i = 1} b_i = 0$?

Answer 1

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.