Dirichlet's test

Let \((a_k)_{k\ge1}\) and \((b_k)_{k\ge1}\) be sequences of real numbers, and let \(A_n\coloneqq\sum_{k=1}^n a_k\). Suppose that

1. the sequence of partial sums \((A_n)_{n\ge1}\) is bounded, and
2. \((b_k)\) is nonincreasing and \(b_k\to0\).

Then the series \(\sum_{k\geq 1} a_k b_k\) converges.