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

Let $M=\max \{\sup A,\sup B\}$. Every element of the union $A\cup B$ is at most $M$, so $M$ is an upper bound. Conversely, every upper bound of $A\cup B$ is an upper bound of each of $A$ and $B$, hence is at least both suprema and therefore at least $M$. Consequently