= Bounded-variation multipliers of a convergent series
If $\sum a_n$ converges and $b_n$ is bounded with $\sum|b_{n+1}-b_n|<\infty$, then $\sum a_nb_n$ converges. For $T_j=\sum_{k=m}^j a_k$, <summation by parts> gives
$$
\left|\sum_{j=m}^n a_jb_j\right|\leq\sup_{j\geq m}|T_j|\left(|b_n|+\sum_{j=m}^{n-1}|b_j-b_{j+1}|\right).
$$
The first factor tends uniformly to zero by the <Cauchy sequence> criterion, and the second has a uniform bound. A bounded <monotone sequence> has finite total variation, so monotone multipliers are included. Eventual monotonicity also suffices, since finitely many initial terms do not affect convergence. Absolute convergence of $\sum a_n$ is unnecessary.
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