1.2 Let $(X, d)$ be a complete metric space and let $F: X \to X$ be such
that $F^N: X \to X$ is a contraction for some positive integer $N$.
Show that $F$ has a unique fixed point $u \in X$ and that for each
$x \in X$, $\lim_{n \to \infty} F^n(x) = u$.
1.3 Using the result obtained in Exercise 1.2, give an alternative proof
for the Picard-Lindelöf theorem (Theorem 1.7).
