= Solution
The quantitative <Picard-Lindelöf theorem> applies on a rectangle $|x-x_0|\leq a$, $\|Z-B\|_\infty\leq b$. Suppose $F$ is continuous there, bounded in <maximum norm> by $M$, and uniformly <Lipschitz continuous> in $Z$. Then the <initial value problem> $Y'=F(x,Y)$, $Y(x_0)=B$, has a unique solution on
$$
\boxed{|x-x_0|\leq h,\qquad h=\min\{a,b/M\},}
$$
with 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.
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