Proposition 3

On every compact interval with left (or right) endpoint t0 contained in J, with length less than r/(M + ε), there exists an approximate solution to within ε of equation (1), with values in S, and equal to x_0 at t_0.*

We suppose that $t_0$ is not the right-hand endpoint of J, and prove the proposition for intervals with left-hand endpoint $t_0$. Let $\mathscr{M}$ be the set of solutions of (1) to within $\varepsilon$, each of which takes values in S, is equal to $x_0$ at $t_0$, and is defined on a half open interval $[t_0, b[$ contained in J (the interval depending on the approximate solution under consideration). First we show that $\mathscr{M}$ is not empty. Let c be right limit of $f(t, x_0)$ at $t_0$; by lemma 1 (IV, p. 165) the function $f(t, x_0 + c(t - t_0))$ has a right limit equal to c at $t_0$, so the restriction of the function $x_0 + c(t - t_0)$ to a sufficiently small half open interval $[t_0, b[$ will belong to $\mathscr{M}$.

We order the set $\mathscr{M}$ by the relation "u is a restriction of v", and show that $\mathscr{M}$ is inductive (Set Theory, III, p. 154). Let $(u_\alpha)$ be a totally ordered subset of $\mathscr{M}$ and $[t_0, b_\alpha[$ the interval where $u_\alpha$ is defined: for $b_\alpha \leq b_\beta$ the function $u_\beta$ is thus an extension of $u_\alpha$. The union of the intervals $[t_0, b_\alpha[$ is an interval $[t_0, b[$ contained in J, and there exists one and only one function u defined on $[t_0, b[$ that coincides with $u_\alpha$ on $[t_0, b_\alpha[$ for each $\alpha$; among the $b_\alpha$ there is an increasing sequence $(b_{\alpha_n})$ tending to b; since u agrees with $u_{\alpha_n}$ on $[t_0, b_{\alpha_n}[$ the function u admits a derivative satisfying (7) at all the points of the complement of a countable subset of $[t_0, b[$, and so is the supremum of the set $(u_\alpha)$ in $\mathscr{M}$.

By Zorn's lemma (Set Theory, III, p. 154, th. 2), $\mathscr{M}$ admits a maximal element $u_0$; we shall show that if $[t_0, t_1[$ is the interval where $u_0$ is defined, then either $t_1$ is the right-hand endpoint of J, or else $t_1 - t_0 \geq r/(M + \varepsilon)$. We argue by contradiction, supposing that neither of these conditions holds; first we show that one can extend $u_0$ by continuity at the point $t_1$; in fact, for any s and t in $[t_0, t_1[$,

by the mean value theorem; Cauchy's criterion shows that $u_0$ admits a left limit $x_1 \in S$ at the point $t_1$. Now let $c_1$ be the right limit at $t_1$ of the function $f(t, x_1)$; one has $\|c_1\| \leq M$; the same argument as that at the beginning of the proof shows that one can extend $u_0$ to a half open interval with left-hand endpoint $t_1$ by the function $x_1 + c_1(t - t_1)$, so that the extended function belongs to $\mathscr{M}$, which is absurd. This proves the proposition.

When f is uniformly continuous on $J \times S$ one can prove prop. 3 without using Zorn's lemma (IV, p. 199, exerc. 1a)).