Rudin: Chapter 3, Exercise 8
Prove that if $\sum a_n$ converges, and if $(b_n)$ is monotonic and bounded, then $\sum a_n b_n$ converges.
(I am not sure whether this applies to complex series; Rudin isn't specific.)
This question is answered here:
but I think the lemma Potato used to prove the result is similar enough to the original question to be worth proving as well.
It was: Prove that if $\sum a_n$ has bounded partial sums, and if $(t_n)$ is decreasing and converges to $0$, then $\sum a_n t_n$ converges.