Principle of Completeness and the Bolzano–Weierstrass Theorem

Why Completeness is Important

The completeness property of the real numbers is the deepest of the axioms that distinguish the real numbers from the rational ones. It has several equivalent formulations, each offering a new perspective.

Consider the sequence 1, 1.4, 1.41, 1.414, 1.4142, ... — these are the decimal approximations of √2. Each subsequent term is more precise than the previous. In the rational numbers, this limit does not exist — √2 is irrational. But we "know" there must be a point, because the real number line is continuous. The completeness property makes this "knowledge" a rigorous mathematical fact.

Four Equivalent Formulations

1. The Supremum Axiom. If a non-empty set is bounded above, then it has a least upper bound (supremum).

2. The Nested Intervals Principle (Cantor). If [a₁, b₁] ⊇ [a₂, b₂] ⊇ ... is a system of nested intervals, then their intersection is non-empty. If the lengths of the intervals tend to zero, the intersection consists exactly of one point.

3. The Bolzano–Weierstrass Theorem. Every bounded sequence has a convergent subsequence.

4. The Cauchy Criterion. A sequence converges if and only if it is fundamental (Cauchy).

All these statements are different faces of the same coin. In the rational numbers, all of them are false.

Bolzano–Weierstrass Theorem: Proof

Let {aₙ} be a bounded sequence: |aₙ| ≤ M for all n.

Method of bisection: the interval [-M, M] contains infinitely many terms of the sequence. Divide it in half — at least one of the half-intervals contains infinitely many terms; choose it. Repeat. We obtain a system of nested intervals whose lengths → 0. By the nested intervals principle, their intersection is exactly one point L.

From each half-interval, choose one term of the sequence with an increasing index — this gives a subsequence converging to L.

Subsequence Limits and the Bolzano–Weierstrass Number

A number to which at least one subsequence converges is called a subsequence limit (or accumulation point) of the sequence.

The greatest subsequence limit is called the upper limit (limsup), the least — the lower limit (liminf). A sequence converges if and only if limsup = liminf.

For the sequence (−1)ⁿ: limsup = 1, liminf = -1. The sequence diverges.

The Nested Intervals Principle in Practice

The nested intervals principle is an algorithm! The method of bisecting an interval to find a root of the equation f(x) = 0 (the bisection method) is a direct application of this principle. We maintain an interval [a, b] on which f changes sign, and at each step we divide it in half, narrowing the search area twofold. The method is guaranteed to converge and is widely used in computational mathematics.

Supremum and Infimum

The supremum (least upper bound) of a set A is the smallest of the numbers greater than or equal to all elements of A. If the maximum of a set exists, then sup A = max A. But the open interval (0, 1) has no maximum, though sup (0,1) = 1.

The infimum (greatest lower bound) is the greatest lower bound.

These concepts are critical throughout analysis. When we say "a function attains its maximum," we are relying on the existence of the supremum.

Compactness and Its Significance

A bounded closed set on the real line is called compact. The Bolzano–Weierstrass theorem states: from any sequence in a compact set, one can extract a convergent subsequence whose limit also belongs to that set.

This property is crucial for extremal problems: a continuous function on a compact set attains its maximum and minimum (Weierstrass theorem). Without compactness, this result may not hold: the function f(x) = x on the interval (0, 1) attains neither sup = 1 nor inf = 0.

The completeness property is the invisible foundation on which the entire edifice of mathematical analysis stands.

Liouville Numbers and Transcendence

The Bolzano–Weierstrass theorem paves the way to even subtler results. Cantor proved that there are only countably many algebraic numbers (roots of polynomials with integer coefficients), which means that "almost all" real numbers are transcendental (are not roots of any polynomial). Yet concretely proving that a number is transcendental is a nontrivial task. The transcendence of e was proved by Hermite (1873), the transcendence of π by Lindemann (1882), which put an end to the problem of squaring the circle.

The bisection method, which follows directly from the nested intervals principle, is one of the fundamental algorithms of numerical analysis. To find a root f(x) = 0 on [a, b], it is guaranteed to find a solution to any accuracy in O(log(1/ε)) steps. Its practical merits: simplicity of implementation, guaranteed convergence, does not require derivatives.

Question for reflection: The sequence {sin(n)} is bounded (|sin n| ≤ 1). The Bolzano–Weierstrass theorem guarantees a convergent subsequence. Is it possible to explicitly describe all the subsequence limits of this sequence?

The Nested Intervals Principle and Its Consequences

Cantor's lemma on nested intervals: if [a₁, b₁] ⊃ [a₂, b₂] ⊃ ... and (bₙ − aₙ) → 0, there exists a unique point c belonging to all the intervals. This is an equivalent form of the completeness property. The proof of Bolzano–Weierstrass via nested intervals: divide [a, b] in half, choose the half where infinitely many terms of the sequence lie. Repeat — obtain a contracting system, and the unique common point is a subsequence limit. Consequence: the uncountability of the real numbers. If [0,1] were countable, by successive application of the lemma we would obtain a point not on the list — a contradiction.