An iterate of the Picard operator is a contraction

Statement

Let $f$ satisfy the Picard–Lindelöf hypotheses at $(t_0, x_0)$ on $J = [t_{\min}, t_{\max}]$ with constants $a, L, K$. Let $X$ be the set of continuous curves $\gamma : J \to \mathbb{R}^n$ with $\|\gamma(t) - x_0\| \le a$ for each $t \in J$, with the distance $d(\gamma, \delta) = \sup_{t \in J} \|\gamma(t) - \delta(t)\|$. Let $P : X \to X$ be the Picard operator and let $T = \max(t_{\max} - t_0,\ t_0 - t_{\min})$. Then for each integer $m \ge 0$ and all $\gamma, \delta \in X$, $$d\left(P^m \gamma, P^m \delta\right) \le \frac{(K T)^m}{m!}\, d(\gamma, \delta).$$ In particular, there is an $m \ge 1$ such that $P^m$ is a contraction of $X$: $(KT)^m / m! < 1$.