Suppose $(X,d)$ is a compact metric space and $F:X\rar X$ a mapping such that for all $x\neq y$: $d(F(x),F(y)) < d(x,y)$. Then $F$ is not onto.
Since $X\times X$ is compact and $(x,y)\mapsto d(F(x),F(y))$ is continuous there are points $a,b\in X$ such that $d(F(a),F(b))=\sup\{d(F(x),F(y)):x,y\in X\}$. Now by assumtion $d(F(a),F(b))=\colon D < d(a,b)\leq diam(X)$, i.e.
$$
\forall x,y\in X:\quad
d(F(x),F(y))\leq D < diam(X)
$$
Thus $F$ cannot be onto, for there are points $x_0,y_0\in X$ such that $d(x_0,y_0)=diam(X)$.