1. The problem statement, all variables and given/known data Suppose F is mapping of a nonempty complete metric space into itself, and that F^3 = F o F o F is a contraction (o's denote composition). Show that f has a unique fixed point. 3. The attempt at a solution Isn't this kind of a trick question? Suppose f does not have a unique fixed point. Then there does not exist a unique x such that f(x)=x. Hence there doesn't exist a unique x s.t. f(f(x))=f(x)=x. Hence there doesn't exist a unique x s.t. f(f(f(x)))=f(f(x))=f(x)=x. But this contradicts the assumption that F^3 is a contraction, since any contraction on a complete metric space has a unique fixed point. What am I missing here, this is too easy!