TITLE: Is zero a limit point of the sequence $(\sqrt n \sin n)$
QUESTION [5 upvotes]: Is zero a limit point of the sequence $(\sqrt n \sin n)_n$?

REPLY [11 votes]: 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.