The supremum is the least upper bound of a set , defined as a quantity such that no member of the set exceeds
, but if is any positive quantity,
however small, there is a member that exceeds (Jeffreys and Jeffreys 1988). When it exists (which
is not required by this definition, e.g., does not exist), it is denoted (or sometimes simply for short). The supremum is implemented in the Wolfram
Language as MaxValue[f,
constr, vars].

More formally, the supremum
for a (nonempty)
subset of the affinely
extended real numbers
is the smallest value
such that for all
we have .
Using this definition, always
exists and, in particular, .

Whenever a supremum exists, its value is unique. On the real line, the supremum of a set is the same as the supremum of its set
closure.

Consider the real numbers with their usual order. Then for any set , the supremum exists (in ) if and only if is bounded from above
and nonempty.