To prove the Bolzano-Weierstrass theorem, we will first need two lemmas.

Lemma 1.

All bounded monotone sequences converge.

proof.

Let be a bounded, nondecreasing sequence. Let denote the set . Then let (the supremum of .)

Choose some . Then there is a corresponding such that . Since is nondecreasing, for all , . But is bounded, so we have . But this implies , so .

(The proof for nonincreasing sequences is analogous.)

Lemma 2.

Every sequence has a monotonic subsequence.

proof.

First a definition: call the th term of a sequence dominant if it is greater than every term following it.

For the proof, note that a sequence may have finitely many or infinitely many dominant terms.

First we suppose that has infinitely many dominant terms. Form a subsequence solely of dominant terms of . Then by definition of “dominant”, hence is a decreasing (monotone) subsequence of ().

For the second case, assume that our sequence has only finitely many dominant terms. Select such that is beyond the last dominant term. But since is not dominant, there must be some such that . Select this and call it . However, is still not dominant, so there must be an with , and so on, inductively. The resulting sequence

is monotonic (nondecreasing).

proof of Bolzano-Weierstrass.

The proof of the Bolzano-Weierstrass theorem is now simple: let be a bounded sequence. By Lemma 2 it has a monotonic subsequence. By Lemma 1, the subsequence converges.