Let $b$ be the limit of $\{b_{n}\}$ and let $d_{n}=b_{n}-b$ when $\{b_{n}\}$ is decreasing and $d_{n}=b-b_{n}$ when $\{b_{n}\}$ is increasing. By Dirichlet’s convergence test, $\sum a_{n}d_{n}$ is convergent and so is $\sum a_{n}b_{n}=\sum a_{n}(b\pm d_{n})=b\sum a_{n}\pm\sum a_{n}d_{n}$.