# Is zero a limit point of the sequence $(\sqrt n \sin n)$

Is zero a limit point of the sequence $(\sqrt n \sin n)_n$?

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What do you think? How many terms of the sequence around some small neighbour of $0$ can you find? – fgp May 13 '13 at 12:05
Seems like I didn't understood the question. – Were_cat May 13 '13 at 12:36
Did you get something from an answer below? – Did May 29 '13 at 10:59

## 1 Answer

By Dirichlet's approximation theorem, $\pi$ can be approximated by rationals $p/n$ so that $|n\pi-p|<\frac{1}{n}$. Since sine is 1-Lipschitz, we get$|\sin(p)|<\frac{1}{n}$ from which the result is immediate.

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Which makes one wonder whether the OP will manage to complete the step going from $|\sin(p)|\lt\frac1n$ for some $p$s to $|\sin(n)|\lt\frac{c}n$ for some $n$s. – Did May 29 '13 at 10:58