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Let $(x_n)_{n\in\mathbb N}$ be a non-negative, non-increasing sequence such that $\sum_{n=1}^{\infty} x_n<\infty$. I want to show that $$\lim_{n\to\infty}\{nx_n\}=0.$$ I’ve definitely seen this question asked on this forum before, but I’m unable to find it now. Any reference or hint would be appreciated.

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marked as duplicate by triple_sec, Pragabhava, colormegone, user228113, Community Apr 8 '16 at 23:19

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