D: HALP GAIZ!

(Sorry if I'm not supposed to post homework here or something.)

My Analysis prof gives hard problems as extra-credit sometimes, and he gave this one during our first class of the semester yesterday.  I can't get anywhere with it. :(  It looks like it should be totally easy, but I'm not seeing it.  I just have to prove that the sequence {sin(1), sin(4), sin(9), ...} doesn't converge. The numbers 1,4,9... are the integer squares.

Anyone have any ideas?

>>7
>But I aint doing it for you

Translation: You can't figure out how to make it work. :P

It's true that for any real \alpha there are arbitrarily large integers q such that for some integer p,  \left|q\alpha-p\right| < 1/q, so the sequence q\alpha comes within \epsilon of an integer infinitely many times.  The only proofs of this I know of, though, use continued fractions or the box principle, and I doubt they can be adapted to show that q\alpha comes close to square integers infinitely many times.  If you know how, I'd be interested to see it.