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.

Notice that:
$$\left(\sum_{n=0}^{+\infty}|a_n|\right)\cdot\left(\sum_{n=0}^{+\infty}|b_n|\right)=\sum_{m,n}|a_m b_n|$$
is finite and our sum is:
$$\sum_{\substack{m,n\\ m\leq n}}|a_m b_n|\leq \sum_{m,n}|a_m b_n|.$$
