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, $\sum a_nd_n$ is convergent and so is $\sum a_nb_n = \sum a_n(b\pm d_n) = b\sum a_n \pm \sum a_nd_n$ .