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?



Convergent subsequence of $\sin n$
As to the convergenge to zero: note that the convergents of the [continuous fraction for $\pi$](http://oeis.org/A001203) provide a rational approximation $|\pi - p\_n/q\_n| < 1/q\_nq\_{n+1}$ so that $\sin p\_n\to 0$.


The sequence of numerators $0, 1, 3, 22, 333, 355,\dots$ is OEIS' [A002485](http://oeis.org/A002485).


