maximum and minimum

The maximum of a set is its single biggest member, and the minimum its single smallest member, provided such a champion actually belongs to the set. A maximum is a supremum that is lucky enough to be attained; a minimum is an attained infimum.

Precisely, M is the maximum of S if M belongs to S and x is at most M for every x in S; the minimum is defined the mirror way. The key contrast with sup and inf is membership: the supremum is the least upper bound whether or not it lies in the set, while a maximum must be an actual element. So a set always could fail to have a maximum even when its supremum exists.

The open interval from 0 to 1 illustrates the gap: its supremum is 1 and its infimum is 0, but it has neither a maximum nor a minimum, because neither endpoint is included. By contrast the closed interval from 0 to 1 has maximum 1 and minimum 0.

Existence of extrema is a genuine theorem in important cases. The extreme value theorem guarantees that a continuous function on a closed bounded interval attains both a maximum and a minimum, a fact that rests squarely on completeness and would fail on the rationals.