I encountered a problem today to prove that $\cos (n!)$ does not have a limit. I have no idea how to do it formally. Could someone help? The simpler the proof (by that I mean less complex theorems are used) the better.

This question appears to be off-topic. The users who voted to close gave this specific reason:

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The question boils down to whether the sequence $cn!$ tends to a limit mod 1, where $c=1/(2\pi)$. There are transcendental numbers $c$ for which the sequence DOES tend to a limit mod 1, so we have to use something about $\pi$. I'm sorry to see the question closed
– SJRNov 2 '10 at 22:26

IMO it would have made sense for people to participate in the meta thread rather than to have an close/open tug-of-war with no discussion. Oh well.
– Ryan BudneyNov 3 '10 at 5:31