If $\sum|a_n|<\infty$ and $\sum |b_n|<\infty$, is the sum $\sum_{k=0}^\infty \sum_{n=0}^\infty |a_n||b_{n+k}|<\infty$? Suppose that $(a_n)_{n=0}^\infty$ and $(b_n)_{n=0}^\infty$ are complex numbers satisfying $\sum_{n=0}^\infty |a_n|<\infty$ and $\sum_{n=0}^\infty |b_n|<\infty$. Is the sum $\sum_{k=0}^\infty  \Big(\sum_{n=0}^\infty |a_n|\cdot |b_{n+k}|\Big)<\infty$? If not, are there further conditions on the $b_n$ to make the sum finite?
I don't have mathematica handy to test a few examples, and I don't have much of an intuition as to what the answer should be. Any help would be greatly appreciated. Thank you.

Yes, the sum is finite:
$$
\sum_{k=0}^\infty \bigg( \sum_{n=0}^\infty |a_n| |b_{n+k}| \bigg) = \sum_{n=0}^\infty |a_n| \sum_{k=0}^\infty |b_{n+k}| \le \sum_{n=0}^\infty |a_n| \sum_{j=0}^\infty |b_j| = \bigg( \sum_{n=0}^\infty |a_n| \bigg) \bigg( \sum_{j=0}^\infty |b_j| \bigg).
$$
The rearrangement in the first step is justified because all terms are nonnegative.
