Let $X$ be compact and suppose $f:X\to X$ satisfies $d(f(x),f(y))<d(x,y)$ whenever $x\neq y$. Show that $f$ has a fixed point.
$X$ compact and $f$ continuous $\rightarrow f(X)$ compact $\rightarrow f(X)\times X$ compact
we know that,
$d:f(X)\times X \to R$ defined as $(f(x),y)\mapsto d(f(x),y)$ is continuous,
so the image of $f(X)\times X$ is a compact subset of $R$
hence the set $F=\{d(f(x),y)|x,y\in X\}$ is a compact subset of $R$ and therefore contains its infimum, say $\alpha = inf F$
so $\exists x_0, y_0 \in X$ such that $\alpha = d(f(x_0),y_0)$
Now how can I show $\alpha=0$ and $x_0=y_0$ ? or is this not the right approach?
