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 Incoming links: Iterated extrema of an array

Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics II / 1 / b / i / Solution 2026-10-25

True. Write

s = sup A

and

u = sup {sup R_{m} : m \geq 1}

. All these suprema exist in

[0, 1]

. Since

R_{m} \subseteq A

, we have

sup R_{m} \leq s

for every

m

, hence

u \leq s

. Conversely, every

x_{m, n} \leq sup R_{m} \leq u

, so

u

is an upper bound of

A

and

s \leq u

. Thus the iterated extrema of an array satisfy

sup A = m \geq 1 sup n \geq 1 sup x_{m, n} .

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