The Bolzano-Weierstrass Theorem states that any bounded sequence of real numbers has a convergent subsequence.

The proof of this theorem is as follows. The Monotone convergence axiom states that any bounded monotonic sequence of real numbers converges, so it only needs to be shown that any sequence of real numbers has a monotonic subsequence.

Let (an) be a sequence of real numbers. Say that k is a maximal index if ak>=al for all l>=k. Then there are two possibilities:

  • There are infinitely many maximal indices k1<k2<k3<...
    Then ak_1>=ak_2>=ak_3>... so (an) has a decreasing subsequence.
  • There are finitely many maximal indices. Let b be an upper bound for the set of maximal indices. Then if n>b, n isn't a maximal index. It can be shown by induction that there exists an increasing subsequence (ak_j). Let k1=b+1. k1 isn't a maximal index, so there exists k2>k1 with ak_2>ak_1. Now, suppose we have found kj. kj isn't a maximal index, so there exists kj+1>kj with ak_j+1>ak_j. So one can find an increasing subsequence.
So in either case, there is a monotonic subsequence.

QED, as they don't say.