Prove that if $\sum a_n$ converges, and if $(b_n)$ is monotonic and bounded, then $\sum a_n b_n$ converges

Question

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:

Little Rudin series convergence exercise ,

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.

Answer 1

The aforementioned lemma is proved as Theorem $3.42$ in Rudin's "Principles of Mathematical Analysis".