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

True. Write $s=\sup A$ and $u=\sup \{\sup R_m:m\ge 1\}$. All these suprema exist in $[0,1]$. Since $R_m\subseteq A$, we have $\sup R_m\le s$ for every $m$, hence $u\le s$. Conversely, every $x_{m,n}\le \sup R_m\le u$, so $u$ is an upper bound of $A$ and $s\le u$. Thus the iterated extrema of an array satisfy