I'll write it formally: Let f(1,x) = sin(x) Let f(n+1,x)=sin(f(n,x)), What is the limit as n -> infinity?

It's fairly easy to prove that the absolute value of f(n,x) is decreasing for every constant x as n is increasing, so the limit exists. From what I calculated, it seems like f(infinity,x)= 0 for all x. Does someone have a proof for that?