proof of Abel’s test for convergence

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 $\sum a_n{b}_n=\sum a_n(b\pm d_n)=b\sum a_n\pm \sum a_n{d}_n$.