TITLE: Convergent subsequence of $\sin n$
QUESTION [9 upvotes]: It is well known (not to me -- ed.) that for every real number $\theta \in [0, 1]$  there exists a sequence $(k_i)$ such that $\lim\sin k_i = \theta,$ but there appear to be no explicit such (infinite) sequences, even for $\theta=0.$ Does anyone know of such?

REPLY [5 votes]: Since $\pi$ is irrational, the sequence $n\bmod 2\pi$ is dense in $[0,2\pi]$. Now use the fact that $\sin$ is continuous.