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?
Best Answer
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 $m,n\to\infty$.