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Least-upper-bound property

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Least-upper-bound property

In mathematics, the least-upper-bound property (sometimes supremum property of the real numbers) is a fundamental property of the real numbers and certain other ordered sets. The property states that any non-empty set of real numbers that has an upper bound necessarily has a least upper bound (or supremum).

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 (see least upper bound axiom), 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 S be a non-empty set of real numbers.

  • A real number x is called an upper bound for S if xs for all sS.
  • A real number x is the least upper bound (or supremum) for S if x is an upper bound for S and xy for every upper bound y of S.

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

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

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

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

has an upper bound in Q, but does not have a least upper bound in Q (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 S be a nonempty set of real numbers, and suppose that S has an upper bound B1. Since S is nonempty, there exists a real number A1 that is not an upper bound for S. Define sequences A1, A2, A3, ... and B1, B2, B3, ... recursively as follows:

  1. Check whether (An + Bn) ⁄ 2 is an upper bound for S.
  2. If it is, let An+1 = An and let Bn+1 = (An + Bn) ⁄ 2.
  3. Otherwise there must be an element s in S so that s>(An + Bn) ⁄ 2. Let An+1 = s and let Bn+1 = Bn.

Then A1A2A3 ≤ ⋯ ≤ B3B2B1 and |AnBn| → 0 as n → ∞. It follows that both sequences are Cauchy and have the same limit L, which must be the least upper bound for S.

Applications

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

Intermediate value theorem

Let f : [a, b] → R be a continuous function, and suppose that f (a) < 0 and f (b) > 0. In this case, the intermediate value theorem states that f must have a root in the interval [a, b]. This theorem can proved by considering the set

S  =  {s ∈ [a, b]  :  f (x) < 0 for all xs} .

That is, S is the initial segment of [a, b] that takes negative values under f. Then b is an upper bound for S, and the least upper bound must be a root of f.

Bolzano–Weierstrass theorem

The Bolzano–Weierstrass theorem for R states that every sequence xn of real numbers in a closed interval [a, b] must have a convergent subsequence. This theorem can be proved by considering the set

S  =  {s ∈ [a, b]  :  sxn for infinitely many n} .

Clearly b is an upper bound for S, so S has a least upper bound c. Then c must be a limit point of the sequence xn, and it follows that xn has a subsequence that converges to c.

Extreme value theorem

Let f : [a, b] → R be a continuous function and let M = sup f ([a, b]), where M = ∞ if f ([a, b]) has no upper bound. The extreme value theorem states that M is finite and f (c) = M for some c ∈ [a, b]. This can be proved by considering the set

S  =  {s ∈ [a, b]  :  sup f ([s, b]) = M} .

If c is the least upper bound of this set, then it follows from continuity that f (c) = M.

Heine–Borel theorem

Let [a, b] be a closed interval in R, and let {Uα} be a collection of open sets that covers [a, b]. Then the Heine–Borel theorem states that some finite subcollection of {Uα} covers [a, b] as well. This statement can be proved by considering the set

S  =  {s ∈ [a, b]  :  [a, s] can be covered by finitely many Uα} .

This set must have a least upper bound c. But c is itself an element of some open set Uα, and it follows that [a, c + δ] can be covered by finitely many Uα for some sufficiently small δ > 0. This proves that c + δS, and it also yields a contradiction unless c = b.

See also

References

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