Theorem 1 (Cauchy)

*Let $\mathbf{f}$ be a Lipschitz function on $I \times H$, let J be a compact subinterval of $I$, not reducing to a single point, $t_0$ a point of J, S an open ball with centre $\mathbf{x}_0$ and radius r, contained in H, and M the least upper bound of $\| \mathbf{f}(t, \mathbf{x}) \|$ on $J \times S$. In these circumstances, for every compact interval K that does not reduce to a single point and is contained in the intersection of J with $]t_0 - r/M, t_0 + r/M[$, and contains $t_0$, there exists one and only one solution of the differential equation $\mathbf{x}' = \mathbf{f}(t, \mathbf{x})$ defined on K, with values in S, and equal to $\mathbf{x}_0$ at the point $t_0$.

Indeed, for $\varepsilon > 0$ sufficiently small, the set $F_\varepsilon$ of approximate solutions of (1) to within $\varepsilon$, defined on K, with values in S, and equal to $\mathbf{x}_0$ at the point $t_0$, is not empty (IV, p. 166, prop. 3); further, if $\mathbf{u}$ and $\mathbf{v}$ belong to $F_\varepsilon$ then, by (15) (IV. p. 170)

for all $t \in K$, so the sets $F_\varepsilon$ form a filter base $\mathscr{G}$ which converges uniformly on $K$ to a continuous function $w$, equal to $x_0$ at $t_0$; also $w$ takes values in $S$, since, for $\varepsilon$ small enough, the functions $u \in F_\varepsilon$ take their values in a closed ball contained in $S$. Since $f(t, u(t))$ tends uniformly on $K$ to $f(t, w(t))$ along $\mathscr{G}$, $w$ satisfies equation (6) of IV, p. 165, so is a solution of (1). The uniqueness of the solution follows immediately from inequality (15) of IV, p. 170 where one takes $\varepsilon_1 = \varepsilon_2 = 0$ and $u(t_0) = v(t_0)$.

We shall say that a function $f$ defined on $I \times H$ is locally Lipschitz if, for every point $(t, x)$ of $I \times H$, there exists a neighbourhood $V$ of $t$ (with respect to $I$) and a neighbourhood $S$ of $x$ such that $f$ is Lipschitz on $V \times S$ (for a constant $k$ depending on $V$ and $S$). By the Borel-Lebesgue theorem, for every compact interval $J \subset I$ and every point $x_0 \in H$ there exists an open ball $S$ with centre $x_0$, contained in $H$, such that $f$ is Lipschitz on $J \times S$; thus $f$ satisfies the hypotheses of lemma 1 of IV, p. 3. When $f$ is locally Lipschitz on $I \times H$ we shall say that the equation $x' = f(t, x)$ is locally Lipschitz on $I \times H$.

We shall generalize and clarify th. 1 of IV, p. 10 for locally Lipschitz equations; we restrict ourselves to the case where $t_0$ is the left endpoint of the interval $I$; one can pass easily to the case where $t_0$ is an arbitrary point of $I$ (cf. IV, §IV ??, p. 9, corollary).