Proof

By hypothesis, the infinite series $\sum_{k=1}^\infty a_k$ is convergent and the sequence $(b_k)_{k\in\mathbb N}$ is bounded and monotonic.
Thus, the sequence $(A_k)_{n\in\mathbb N}$ of partial sums $A_k:=\sum_{j=1}^k a_j$ is convergent.
Moreover, since every monotonic bounded sequence is convergent, $(b_k)_{k\in\mathbb N}$ is convergent.
Therefore, the telescoping series $\sum_{k=1}^\infty (b_k-b_{k+1})$ is convergent, it is even absolutely convergent, since all of its terms are either $\ge 0$ or $\le 0.$
Since $A_k$ are bounded, the series $\sum_{k=1}^\infty A_k(b_k-b_{k+1})$ is convergent and the series $\sum_{k=1}^\infty A_kb_{k+1}$ is convergent.
By the Abel's lemma, the series $\sum_{k=1}^\infty a_kb_k$ is convergent.
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Heuser Harro: "Lehrbuch der Analysis, Teil 1", B.G. Teubner Stuttgart, 1994, 11th Edition