 Solution

A Lipschitz condition here is uniform in the independent variable: there is a finite constant $L\ge 0$ such that, whenever both $(x,Z)$ and $(x,W)$ belong to the cube,The same $L$ must work for every allowed $x,Z,W$. Part (c) proves this with $L=1$. This controls changes in the dependent variables; no Lipschitz condition in $x$ is required for the Picard-Lindelöf theorem.