Let $(a_n)_{n\in \mathbb N}$ be a sequence of real numbers with $a_n\ge0$ for all $n\in\mathbb N$ and suppose $\sum_{n=1}^\infty a_n$ converges. Show that if $(b_n)_{n\in\mathbb N}$ is any bounded sequence of real numbers, then $\sum_{n=1}^\infty a_n b_n$ converges absolutely.
The Monotone Convergence Theorem is an obvious choice here. I think I can see the sequence of partial sums is bounded, but how would I show that the it is monotonically increase or decreasing without anymore information on $(b_n)$? Should I divide it into cases?
Thanks.
Best Answer
Since $b_n$ is bounded, there exists some $M$ such that $|b_n| \leq M$ for all $n$. Try comparing the non-negative series $\sum_n a_n |b_n|$ and $\sum_n M a_n$.