Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Discrete mathematics, continuous mathematics, and probability / 3 / c / ii / Solution

An element $m$ is a maximum of $S$ if $m\in S$ and $s\le m$ for every $s\in S$. Thus it is an upper bound that is itself a member. Every maximum is the least upper bound: any upper bound must lie above $m$ because $m\in S$. A supremum outside $S$ need not be a maximum.