Least-upper-bound property

In mathematics, the least-upper-bound property (sometimes called completeness or supremum property or l.u.b. property)[1] is a fundamental property of the real numbers. More generally, a partially ordered set X has the least-upper-bound property if every non-empty subset of X with an upper bound has a least upper bound (supremum) in X. Not every (partially) ordered set has the least upper bound property. For example, the set {\displaystyle \mathbb {Q} }\mathbb {Q} of all rational numbers with its natural order does not have the least upper bound property.

The least-upper-bound property is one form of the completeness axiom for the real numbers, and is sometimes referred to as Dedekind completeness.[2] It can be used to prove many of the fundamental results of real analysis, such as the intermediate value theorem, the Bolzano–Weierstrass theorem, the extreme value theorem, and the Heine–Borel theorem. It is usually taken as an axiom in synthetic constructions of the real numbers, and it is also intimately related to the construction of the real numbers using Dedekind cuts.

In order theory, this property can be generalized to a notion of completeness for any partially ordered set. A linearly ordered set that is dense and has the least upper bound property is called a linear continuum.

Every non-empty subset {\displaystyle M}M of the real numbers {\displaystyle \mathbb {R} }\mathbb {R} which is bounded from above has a least upper bound.

1 Statement of the property

1.1 Statement for real numbers

1.2 Generalization to ordered sets

2 Proof

2.1 Logical status

2.2 Proof using Cauchy sequences

3 Applications

3.1 Intermediate value theorem

3.2 Bolzano–Weierstrass theorem

3.3 Extreme value theorem

3.4 Heine–Borel theorem

4 History

5 See also