Let $(X,d)$ be nonempty compact metric space and $f : X\to X$ be a function satisfying $d(f (x), f (y)) < d(x, y)$ for all distinct pair of points $x, x \in X$. Show that $f$ have a fixed point and this fixed point is unique.
Hint: Define the function $g : X \to R$ by $g(x) = d(f (x), x)$. Assume that $f (x) \neq x$ for all $x \in X$. Use compactness to show the existence of a point $a \in X$ such that $g(a) \leq g (x)$ for all $x \in X$. Deduce a contradiction by considering $x=f (a)$.