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For a nonempty bounded array of real numbers, taking the supremum of each row and then the supremum of those values gives the supremum of all entries. Every row supremum is bounded by the global supremum; conversely, every entry is bounded by the iterated supremum. The same argument with lower bounds proves ${\inf }_m{\inf }_n{x}_{m,n}={\inf }_{m,n}{x}_{m,n}$. Mixed infima and suprema satisfy the weak minimax inequality rather than an equality in general.

True. For every $m,n$,Fix $m$. The number $\sup R_m$ is an upper bound for every column infimum, so ${\sup }_n\inf C_n\le \sup R_m$. This holds for every $m$, making ${\sup }_n\inf C_n$ a lower bound of the row suprema. Taking their infimum gives the weak minimax inequality