Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics II / 1 / b / iii

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- Solution iii

Solution

True. For every

m, n

,

in f C_{n} \leq x_{m, n} \leq sup R_{m} .

Fix

m

sup R_{m}

is an upper bound for every column infimum, so

sup_{n} in f C_{n} \leq sup R_{m}

. This holds for every

m

, making

sup_{n} in f C_{n}

a lower bound of the row suprema. Taking their infimum gives the weak minimax inequality

n \geq 1 sup m \geq 1 in f x_{m, n} \leq m \geq 1 in f n \geq 1 sup x_{m, n} .

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