(adsbygoogle = window.adsbygoogle || []).push({}); 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!

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# Analysis Comp question: function that's a contraction mapping when composed w/itself

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