Problem 1532 (difficulty: 6/10)

Let \(\displaystyle a_1,a_2,\ldots\) be a decreasing sequence of positive numbers that converges to \(\displaystyle 0\), and let \(\displaystyle b_1,b_2,\ldots\) be a sequence of complex numbers such that the partial sums \(\displaystyle b_1+\ldots+b_n\) are bounded by a constant independent of \(\displaystyle n\). Prove that \(\displaystyle \displaystyle\sum_{n=1}^\infty a_nb_n\) is convergent.

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