How to show that $\lim_{n\to \infty}n\sin(2\pi en!)=2\pi$? [duplicate]

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Please give me some hint to proceed. I'm clueless:

Show that, $\lim\limits_{n\to \infty}n\sin(2\pi en!)=2\pi$

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marked as duplicate by Martin Sleziak, Marc van Leeuwen, Davide Giraudo, drhab, Sami Ben Romdhane2 days ago

Note that $e=\sum_{k=0}^\infty\frac1{k!}$, hence $en!=\sum_{k=0}^n\frac{n!}{k!}+\frac1{n+1}+\ldots$ is quite close to an integer.