$a_n$ is convergent, $b_n$ bounded, prove $\sum a_n b_n$ converges

Question

Suppose that $\sum_{n=0}^\infty a_n$ is a convergent series with $a_n$ > $0$ and suppose that $(b_n){_{n\in\mathbb{N}}}$ is a bounded sequence of positive numbers. Show that $\sum_{n=0}^\infty a_n b_n$ is convergent.

Since $b_n$ is bounded and $b_n$ > $0$, can we conclude that there exists M > $0$ such that $0$ < $b_n$ < M or only that $b_n$ > $0$? If we cannot conclude that $b_n$ < M then how can we answer this question?

Answer 1

Note that $$ \left|\sum_{k=n+1}^ma_kb_k\right|\leq\sum_{k=n+1}^m|a_kb_k|\leq M \sum_{k=n+1}^m|a_k|\to0 $$ as $m,n\to\infty$.

Answer 2

Since $(b_n)$ is bounded and positive, there is an $M\geq b_n >0$ for alle $n\in\mathbb{N}$.By the direct comparision test we have:

$\sum_{n=0}^\infty a_nb_n\leq \sum_{n=0}^\infty a_n\cdot M=M\underbrace{\sum_{n=0}^\infty a_n}_{<\infty}<\infty$