Least-upper-bound property

Every non-empty subset $M$ of the real numbers $ℝ$ which is bounded from above has a least upper bound.

In mathematics, the least-upper-bound property (sometimes called completeness, supremum property or l.u.b. property) is a fundamental property of the real numbers. More generally, a partially ordered set Template:Math has the least-upper-bound property if every non-empty subset of Template:Math with an upper bound has a least upper bound (supremum) in Template:Math. Not every (partially) ordered set has the least upper bound property. For example, the set $\mathbb{Q}$ 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. 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.

Statement of the property

Statement for real numbers

Let Template:Math be a non-empty set of real numbers.

• A real number Template:Math is called an upper bound for Template:Math if Template:Math for all Template:Math.
• A real number Template:Math is the least upper bound (or supremum) for Template:Math if Template:Math is an upper bound for Template:Math and Template:Math for every upper bound Template:Math of Template:Math.

The least-upper-bound property states that any non-empty set of real numbers that has an upper bound must have a least upper bound in real numbers.

Generalization to ordered sets

Red: the set $\{x\in 𝐐:x^2\le 2\}$. Blue: the set of its upper bounds in $𝐐$.

Template:Main article More generally, one may define upper bound and least upper bound for any subset of a partially ordered set Template:Math, with “real number” replaced by “element of Template:Math”. In this case, we say that Template:Math has the least-upper-bound property if every non-empty subset of Template:Math with an upper bound has a least upper bound in Template:Math.

For example, the set Template:Math of rational numbers does not have the least-upper-bound property under the usual order. For instance, the set

$\{x\in 𝐐:x^2\le 2\}=𝐐\cap (-\sqrt{2},\sqrt{2})$

has an upper bound in Template:Math, but does not have a least upper bound in Template:Math (since the square root of two is irrational). The construction of the real numbers using Dedekind cuts takes advantage of this failure by defining the irrational numbers as the least upper bounds of certain subsets of the rationals.

Proof

Logical status

The least-upper-bound property is equivalent to other forms of the completeness axiom, such as the convergence of Cauchy sequences or the nested intervals theorem. The logical status of the property depends on the construction of the real numbers used: in the synthetic approach, the property is usually taken as an axiom for the real numbers (see least upper bound axiom); in a constructive approach, the property must be proved as a theorem, either directly from the construction or as a consequence of some other form of completeness.

Proof using Cauchy sequences

It is possible to prove the least-upper-bound property using the assumption that every Cauchy sequence of real numbers converges. Let Template:Math be a nonempty set of real numbers. If Template:Math has exactly one element, then its only element is a least upper bound. So consider Template:Math with more than one element, and suppose that Template:Math has an upper bound Template:Math. Since Template:Math is nonempty and has more than one element, there exists a real number Template:Math that is not an upper bound for Template:Math. Define sequences Template:Math and Template:Math recursively as follows:

1. Check whether Template:Math is an upper bound for Template:Math.
2. If it is, let Template:Math and let Template:Math.
3. Otherwise there must be an element Template:Math in Template:Math so that Template:Math. Let Template:Math and let Template:Math.

Then Template:Math and Template:Math as Template:Math. It follows that both sequences are Cauchy and have the same limit Template:Math, which must be the least upper bound for Template:Math.

Applications

The least-upper-bound property of Template:Math can be used to prove many of the main foundational theorems in real analysis.

Intermediate value theorem

Let Template:Math be a continuous function, and suppose that Template:Math and Template:Math. In this case, the intermediate value theorem states that Template:Math must have a root in the interval Template:Math. This theorem can be proved by considering the set

Template:Math.

That is, Template:Math is the initial segment of Template:Math that takes negative values under Template:Math. Then Template:Math is an upper bound for Template:Math, and the least upper bound must be a root of Template:Math.

Bolzano–Weierstrass theorem

The Bolzano–Weierstrass theorem for Template:Math states that every sequence Template:Math of real numbers in a closed interval Template:Math must have a convergent subsequence. This theorem can be proved by considering the set

Template:Math

Clearly, $a\in S$, and Template:Math is not empty. In addition, Template:Math is an upper bound for Template:Math, so Template:Math has a least upper bound Template:Math. Then Template:Math must be a limit point of the sequence Template:Math, and it follows that Template:Math has a subsequence that converges to Template:Math.

Extreme value theorem

Let Template:Math be a continuous function and let Template:Math, where Template:Math if Template:Math has no upper bound. The extreme value theorem states that Template:Math is finite and Template:Math for some Template:Math. This can be proved by considering the set

Template:Math.

By definition of Template:Math, Template:Math, and by its own definition, Template:Math is bounded by Template:Math. If Template:Math is the least upper bound of Template:Math, then it follows from continuity that Template:Math.

Heine–Borel theorem

Let Template:Math be a closed interval in Template:Math, and let Template:Math be a collection of open sets that covers Template:Math. Then the Heine–Borel theorem states that some finite subcollection of Template:Math covers Template:Math as well. This statement can be proved by considering the set

Template:Math.

The set Template:Math obviously contains Template:Math, and is bounded by Template:Math by construction. By the least-upper-bound property, Template:Math has a least upper bound Template:Math. Hence, Template:Math is itself an element of some open set Template:Math, and it follows for Template:Math that Template:Math can be covered by finitely many Template:Math for some sufficiently small Template:Math. This proves that Template:Math and Template:Math is not an upper bound for Template:Math. Consequently, Template:Math.

History

The importance of the least-upper-bound property was first recognized by Bernard Bolzano in his 1817 paper Rein analytischer Beweis des Lehrsatzes dass zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege.

See also

• List of real analysis topics

References

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