= Solution
Use the interior <ordered subsolution and supersolution> correction from part (a). Recursively solve the <Poisson equation> $\Delta u_{k+1}=V(u_k)$ with boundary value $\psi$, starting from $u_0=\varphi^-$. Part (a) gives $u_0\leq u_1\leq\varphi^+$. If $u_{k-1}\leq u_k\leq\varphi^+$, then
$$
\Delta(u_{k+1}-u_k)=V(u_k)-V(u_{k-1})\leq0,\qquad
\Delta(\varphi^+-u_{k+1})\leq V(\varphi^+)-V(u_k)\leq0.
$$
The first difference has zero boundary values, the second nonnegative values. The <weak maximum principle for elliptic operators> applied to their negatives gives $u_k\leq u_{k+1}\leq\varphi^+$. Induction proves the bounded <monotone sequence>
$$
\boxed{\varphi^-\leq u_1\leq u_2\leq\cdots\leq\varphi^+.}
$$
Without interior ordering, the counterexample in part (a) disproves the claim.
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