Suppose that $\sum_{n=0}^\infty a_{n}$ is a convergent series, with $a_{n}\gt0$ and suppose that $b_{n}\gt0$ is a bounded sequence. Then show that the series $\sum_{n=0}^\infty (a_{n}b_{n}$) is convergent.
Take B $\in \mathbb{R}$ with $|b_{n}|\le$ B for a all n. Then |$a_{n}\cdot b_{n}$| $\le$|B$\cdot a_{n}$| for all n. Since $\sum_{n=0}^\infty a_{n}$ is convergent then $\sum_{n=0}^\infty B \cdot a_{n}$ (arithmetic of series). Hence by comparison test $\sum_{n=0}^\infty a_{n} \cdot b_{n}$ is convergent
Is there a better way to do this? maybe using partial sums ??
By the Monotone convergence theorem, for a positive term series, convergence is equivalent to the boundedness of the sequence of partial sums (because it is increasing). It easily seen that the sequence of partial sums for the second series is bounded by the first sequence of partial sums times $B$.