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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.

A real increasing sequence bounded above converges to its supremum; a decreasing sequence bounded below converges to its infimum. For the increasing case, the definition of supremum provides a term above $L-\epsilon$, and every later term remains between that term and $L$. Negation reduces the decreasing case to the increasing case. This elementary result is distinct from the measure-theoretic monotone convergence theorem for integrals.

For a L2-bounded continuous martingale, the terminal and maximal norms satisfyThe first inequality follows from almost sure convergence. Apply the Doob L2 maximal inequality on $[0,T]$ and then the monotone convergence theorem as $T\to \infty$ for the second. These are equivalent norms on the space modulo indistinguishability of stochastic processes; neither inequality needs a zero initial value.

In a probability measure-preserving system whose transformation is an ergodic transformation, a nonnegative measurable $f$ with $\int fd\mu =+\infty$ has $N^{-1}{\sum }_{j=0}^{N-1}f(T^{j}x)\to +\infty$ almost everywhere. Apply the pointwise ergodic theorem to $\min (f,K)$ on a common full-measure set for integer $K$, then bound the lower limit by each truncated integral. The monotone convergence theorem sends those integrals to infinity. The finite-integral case is the ordinary pointwise ergodic theorem.