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.