Let $f$ be a mapping of a metric space $(X, d)$ into itself such that $d(f(x), f(y))<$ $d(x, y)$ for all distinct $x, y$ in $\mathrm{X}$.

Show that $f(x)$ and $d(x, f(x))$ are continuous functions of $x$.

Now suppose that $(X, d)$ is compact and let

$$h=\inf _{x \in X} d(x, f(x))$$

Show that we cannot have $h>0$.

[You may assume that a continuous function on a compact metric space is bounded and attains its bounds.]

Deduce that $f$ possesses a fixed point, and that it is unique.