Bellman comparison with nonnegative rewards and superunit discount ($\beta \ge 1,r\ge 0$)

For nonnegative rewards, a nonnegative value satisfying the maximizing Bellman equation dominates every competing policy, even for $\beta \ge 1$. Iterate the one-step inequality, discard the nonnegative terminal value, then apply the monotone convergence theorem to the accumulated reward.