# A problem with a compact metric space and $f \colon X \to X$ a uniformly continuous function [duplicate]

Let (X,d) be a compact metric space and $$f \colon X \to X$$ a uniformly continuous function so that $$d( f(x),(f(y)) < d(x,y) , \forall x,y \in X , x \neq y$$ . Prove that there exists x' in X so that $$f(x') = x'.$$ I would choose any x and f(x) in X and then look at $$d( f(x),f(f(x))$$ and then $$d( f(f(x)),f(f(f(x)))$$ and so on . Because of the assumtion this would lead to zero. Edit : found the answer this is a duplicate

## marked as duplicate by Robert Z, Saucy O'Path, Paul Frost, user10354138, RebellosNov 17 '18 at 13:58

• $f$ should be a function from $X$ to $X$. – Joppy Nov 17 '18 at 11:12