A
totally ordered set
X is said to have the "least upper bound property" if any subset of
X which is bounded above has a
supremum (or "
least upper bound") in
X.
This is one of the most important properties possessed by the real numbers, and it distinguishes them from the rationals, which do not have this property. It is a theorem that there is a unique ordered field with the least upper bound property, and that this field (which we call the real numbers) contains the rational numbers as a proper subset. (See completeness axiom.) Many important properties of the real numbers (such as the fact that they are a complete field) can be proved by appealing to the least upper bound property.