This is a nice problem on series convergence that I recently stumbled upon. Given a non-negative sequence of real numbers $(a_n)$ such that $$\sum_{n=1}^\infty a_n < \infty,$$ show that there exists a non-decreasing sequence of non-negative numbers $b_n$ such that $$b_n \to \infty \quad\text{ and } \quad \sum_{n=1}^\infty a_n b_n < \infty.$$ In other words, for every convergent series with non-negative terms, there is another convergent series with "substantially larger" terms.
I have a solution (see below), but maybe someone else has a different, simpler, and/or more elegant solution.