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A value function assigns each dynamic programming state and time the best objective function value attainable over the remaining decisions. It is a score, whereas a control policy specifies the actions producing that score. In the 0-1 knapsack problem above, $V_3(5)=7$ is the score, and choosing the items of weights $2$ and $3$ attains it. In sequential mathematical optimization, the Bellman equation computes a value function by comparing immediate reward plus continuation value. Use a supremum or infimum when the best score need not be attained by any admissible control policy.

For a nonempty bounded array of real numbers, each column infimum is at most every row supremum, because ${\inf }_k{x}_{k,n}\le x_{m,n}\le {\sup }_j{x}_{m,j}$ for all $m,n$. Taking the supremum over columns and then the infimum over rows proves the displayed inequality. Equality need not hold: for $x_{m,n}=1_{\{m=n\}}$ on positive integer indices, every column infimum is zero and every row supremum is one. This elementary bound is the order-theoretic starting point for stronger minimax theorems with additional hypotheses.