Corollary (Peano's theorem)

If E is of finite dimension over $\mathbf{R}$ then there exists a solution of (1) with values in S and equal to $x_0$ at $t_0$, on every compact interval K with left (or right) endpoint $t_0$ contained in J and having length $< r/M$.

Indeed, by prop. 3, once $n$ is large enough there is an approximate solution $u_n$ of (1) to within $1/n$, defined on $K$, with values in $S$, and equal to $x_0$ at $t_0$. Further, from some value of $n$ on, $u_n(K)$ is contained in a closed ball with centre $x_0$ and radius $< r$, independent of $n$. The set of $u_n$ is equicontinuous (prop. 4), and since $E$ is finite dimensional the set $S$ is relatively compact in $E$; so for every $t \in K$ the set of $u_n(t)$ is relatively compact in $E$. By Ascoli’s theorem (Gen. Top., X, p. 290, th. 2) the set of $u_n$ is relatively compact in the space $\mathscr{F}(K; E)$ of maps from $K$ into $E$ endowed with the uniform norm. Thus there is a sequence extracted from $(u_{n_k})$ of $(u_n)$ which converges uniformly on $K$ to a continuous function $u$. One has $u(K) \subset S$, so $t \mapsto f(t, u(t))$ is defined on $K$; by lemma 1 (IV, p. 165), $f(t, u_{n_k}(t))$ converges uniformly to $f(t, u(t))$ on $K$; by (IV, p. 4, formula (7)), $u_{n_k}$ is a primitive of a function which tends uniformly to $f(t, u(t))$ on $K$, so (II, p. 52, th. 1) $u$ is a solution of (1) on $K$, and equal to $x_0$ at the point $t_0$.