A totally ordered
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.