Let $(X,d)$ be a compact metric space and $f : X \to X$ be isometric, i.e. for every $x,y \in X$ : $ d(f(x),f(y)) = d(x,y) $. How I can show the following? $f(X) = X$
My first thought was to show the surjection $ x \in X \setminus f(x) $. Is this the right approach?
Thanks in advance
