 Solution

The quantitative Picard-Lindelöf theorem applies on a rectangle $|x-x_0|\le a$, $\Vert Z-B{\Vert }_{\infty }\le b$. Suppose $F$ is continuous there, bounded in maximum norm by $M$, and uniformly Lipschitz continuous in $Z$. Then the initial value problem $Y^{\prime }=F(x,Y)$, $Y(x_0)=B$, has a unique solution onwith the usual interpretation $b/M=\infty$ if $M=0$. The solution remains in the dependent-variable ball. Its endpoints are included when $F$ is continuous on the closed rectangle, as here. Equivalently, existence first on every smaller interval and the derivative bound give extension to those endpoints.

For the given cube, $x_0=0$ and $a=b=1$. Thus the theorem gives $h=\min \{1,1/M\}$ once a bound $M$ and a uniform Lipschitz constant have been established. The theorem does not require $Lh<1$: convergence of Picard iteration can be proved by factorial estimates or by using successive shorter intervals.