Existence of unique fixed point in compact Metric space [duplicate]

Let $(X,d)$ be compact. Show: for a map $f$ that when $\forall x, y \in X$ with $x\neq y$

$d(f(x),f(y))<d(x,y)$ is fulfilled.

Then $f$ has a unique fixed point.

Assume there are two fixed points, let $x$ and $y$.
As $f(x)=x$ and $f(y)=y$, then $$d(f(x),f(y))=d(x,y)<d(x,y),$$ a contradiction.
• The statement $f$ has a unique fixed point means $f$ has a fixed point and it does not have more than one fixed point. Where did you prove the existence of a fixed point? – Kavi Rama Murthy May 22 '18 at 8:58