The Bolzano-Weierstrass Theorem

2 The Bolzano-Weierstrass theorem

We have seen that every convergent sequence is bounded, but that not every bounded sequence is convergent. For example, (-1)n defines a non-convergent bounded sequence. Thus there is no full converse to the statement that every convergent sequence is bounded. The Bolzano-Weierstrass theorem, to be presented next, gives a partial converse.

Theorem 2.1 (Bolzano-Weierstrass)

Suppose that (an) is a bounded sequence of real numbers. Then (an) has a convergent subsequence.

Proof

Suppose that (an) is bounded. For the purposes of this proof, say that the nth term an of the sequence is dominant if an≥ak for all k≥n. Our proof now splits into two cases.

Case 1. Suppose that (an) has infinitely many dominant terms

an1,an2,an3,an4,…

where n1<n2<n3<…. Then by the definition of dominant we have an1≥an2≥an3≥… and this gives a bounded monotonic nonincreasing subsequence which converges by the monotone convergence theorem.

Case 2. If not, then the sequence (an) has only finitely many dominant terms. Choose n1 beyond the last dominant term. (So, for example, we may take the last dominant term ak, and define n1=k+1.) As an1 is not dominant there is n2>n1 such that an1<an2, and as an2 is not dominant there is n3>n2 such that an2<an3, and so on. Continuing in this way we obtain a monotonic bounded increasing subsequence an1<an2<an3<… which once again converges by the monotone convergence theorem.

The Bolzano-Weierstrass theorem is an important and powerful result related to the so-called compactness of intervals [a,b] in the real numbers, and you may well see it discussed further in a course on metric spaces or topological spaces. Note that the completeness of the reals (in the form of the monotone convergence theorem) is an essential ingredient of the proof. We shall apply it in another section of these notes to give an alternative view of the completeness of ℝ.

The Bolzano-Weierstrass Theorem

1 Introduction

2 The Bolzano-Weierstrass theorem