Least-upper-bound property

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.

It is possible to prove the least-upper-bound property using the assumption that every Cauchy sequence of real numbers converges.

Since S is nonempty and has more than one element, 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: Then A1 ≤ A2 ≤ A3 ≤ ⋯ ≤ B3 ≤ B2 ≤ B1 and |An − Bn| → 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. The least-upper-bound property of R can be used to prove many of the main foundational theorems in real analysis.

In this case, the intermediate value theorem states that f must have a root in the interval [a, b].

This theorem can be proved by considering the set 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. 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.

If c is the least upper bound of S, then it follows from continuity that f (c) = M. Let [a, b] be a closed interval in R, and let {Uα} be a collection of open sets that covers [a, b].

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.