= Weak minimax inequality
{title2=$\sup_n\inf_m x_{m,n}\leq\inf_m\sup_n x_{m,n}$}
For a nonempty bounded array of <real numbers>, each column <infimum> is at most every row <supremum>, because $\inf_kx_{k,n}\leq x_{m,n}\leq\sup_jx_{m,j}$ for all $m,n$. Taking the <supremum> over columns and then the <infimum> over rows proves the displayed inequality. Equality need not hold: for $x_{m,n}=\mathbf1_{\{m=n\}}$ 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.
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