Let X be a subset of the real numbers.

A number a is an upper bound for X if x<=a, for all x in X.

Note that there are subsets that don't have an upper bound, for example Z.

A real number A is called the supremum of X if it is an upper bound of X and if whenever a is another upper bound of X then A <= a. The usual notation is sup X. (Can you figure out why if a supremum exists it is unique?)

By convention, when X has no upper bound we write sup X=infinity.

Note that the supremum is a different concept from the maximum. For example if we consider the set

X={1-1/x : x is a positive integer}

so that X={0,1/2,2/3,3/4,...} then we can see that sup X=1 but 1 is not actually a member of X so it is not a maximum.

Obviously the supremum can be defined in more general contexts than subsets of the real line. But this suffices to explain the idea. A simple but important fact about the real numbers is that if a nonempty subset of R has an upper bound then it has a finite supremum. This fact is a defining axiom of the real numbers (the completeness property).

See also infimum.