What are the limit points of $\sin(2n \pi/3)$? I want to find the limit points of $a_n = \sin(2n \pi/3)$. So far I've identified the subsequences $a_{3n} = 0$ and $a_{1} = \frac{\sqrt{3}}{2}$ and $a_2 = -\frac{\sqrt{3}}{2}$ so are the limit points $\left\{0, \pm \frac{\sqrt{3}}{2}\right\}$? How do I show there aren't more?

A limit point is a point (number) for which there is a subsequence that converges to that point. Now any subsequence must have either infinitely many $0$'s or infinitely many $\dfrac {\sqrt 3} 2$'s or infinitely many $-\dfrac {\sqrt 3} 2$'s, so the limit can ONLY be one of these three numbers.
