Least-upper-bound property

Proof

Logical status

For the real numbers, 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 a synthetic approach, the property is usually taken as an axiom; in other approaches, 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

The least-upper-bound property can be proved from the Archimedean property together with the assumption that every Cauchy sequence of real numbers converges. Let S be a nonempty set of real numbers. If S has exactly one element, then its only element is a least upper bound. So consider S with more than one element, and suppose that S has an upper bound B₁. 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:

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

Then A₁ ≤ A₂ ≤ A₃ ≤ ⋯ ≤ B₃ ≤ B₂ ≤ B₁. Now |A_n − B_n| → 0 as n → ∞, by the Archimedean property. 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 be proved by considering the set

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

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 x_n 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] : s ≤ x_n for infinitely many n}

Clearly, ${\displaystyle a\in S}$, and S is not empty. In addition, 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 x_n, and it follows that x_n 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} .

By definition of M, a ∈ S, and by its own definition, S is bounded by b. If c is the least upper bound of S, then it follows from continuity that f (c) = M.

Heine–Borel theorem

Let [a, b] be a closed interval in R, and let {U_α} be a collection of open sets that covers [a, b]. Then the Heine–Borel theorem states that some finite subcollection of {U_α} 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 U_α} .

The set S obviously contains a, and is bounded by b by construction. By the least-upper-bound property, S has a least upper bound c ∈ [a, b]. Hence, c is itself an element of some open set U_α, and it follows for c < b that [a, c + δ] can be covered by finitely many U_α for some sufficiently small δ > 0. This proves that c + δ ∈ S and c is not an upper bound for S. Consequently, c = b.

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.