# Supremum

### From Glossary

(abbr. *Sup*). This is the least upper bound on a (real-valued) function over (a subset of) its domain. If is unbounded from above,
and if the domain is empty,
Otherwise, suppose U is any upper bound:
for all
is a *least* upper bound if, for any
there exists in the domain for which
(That is, we can get arbitrarily close to in the range of ) Note that the supremum is always defined, and its range is in the
extended reals. The supremum is the maximum, if it is attained by some point in its domain.