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.