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.