= Solution
{c}
Write $M_0=\sup_xu_0(x)<\infty$ and assume $\lambda(1+pM_0^{p-1})\leq1$. Suprema are understood where the printed maxima on the infinite domain are not attained. We prove the invariant interval $0\leq U_j^m\leq M_0$ simultaneously with the stated speed bound.
Suppose the interval property holds at time $m$. By the <mean value theorem>, the divided difference
$$
a_j^m=\frac{f(U_j^m)-f(U_{j-1}^m)}{U_j^m-U_{j-1}^m}
$$
is between $1$ and $1+pM_0^{p-1}$; when the two states coincide define it as $f'(U_j^m)$. The update is the <convex combination>
$$
U_j^{m+1}=(1-\lambda a_j^m)U_j^m+\lambda a_j^mU_{j-1}^m.
$$
Both coefficients are nonnegative and sum to one, so the same interval property holds at time $m+1$. It is true initially and hence for all times by induction. Therefore
$$
\boxed{\lambda\sup_j|1+p(U_j^m)^{p-1}|\leq\lambda(1+pM_0^{p-1})\leq1\quad\text{for every }m.}
$$
The propagated nonnegativity is essential; it keeps the flux increasing on the entire range of the numerical solution.
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