Solution
= Solution
{c}
The convex-combination formula proved in part (i) gives $0\leq U_j^{m+1}\leq\sup_\ell U_\ell^m$ for every $j$. Taking the supremum and using nonnegativity gives
$$
\boxed{\|U^{m+1}\|_{\ell^\infty}\leq\|U^m\|_{\ell^\infty}\leq\|U^0\|_{\ell^\infty}.}
$$
This is the <discrete maximum principle> for the nonlinear upwind step and proves the requested conditional $\ell^\infty$ stability under the propagated <Courant–Friedrichs–Lewy condition>.