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

True. Set $l=\inf A$ and $v=\inf \{\inf R_m:m\ge 1\}$. Because $R_m\subseteq A$, every row infimum satisfies $\inf R_m\ge l$, so $v\ge l$. But $x_{m,n}\ge \inf R_m\ge v$ for all $m,n$, making $v$ a lower bound of $A$. Therefore $v\le l$, and