Proof

By hypothesis, the sequence of partial sums $(A_n)_{n\in\mathbb N}$ with $A_n:=\sum_{k=0}^n a_k$ is bounded and the sequence $(b_k)_{k\in\mathbb N}$ is monotonic and convergent to zero $\lim_{k\to\infty} b_k=0$.
Since $(A_n)_{n\in\mathbb N}$ is bounded, it follows that $\lim_{k\to\infty} A_kb_{k+1}=0.$
Moreover, since the telescoping series $\sum_{k=1}^\infty (b_k-b_{k+1})$ is convergent, the series $\sum_{k=1}^\infty A_k(b_k-b_{k+1})$ 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