Bolzano-Weierstrass Theorem (thing)

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 (a_n) be a sequence of real numbers. Say that k is a maximal index if a_k>=a_l for all l>=k. Then there are two possibilities:

• There are infinitely many maximal indices k₁<k₂<k₃<...
  Then a_{k_1}>=a_{k_2}>=a_{k_3}>... so (a_n) 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 (a_{k_j}). Let k₁=b+1. k₁ isn't a maximal index, so there exists k₂>k₁ with a_{k_2}>a_{k_1}. Now, suppose we have found k_j. k_j isn't a maximal index, so there exists k_j+1>k_j with a_{k_j+1}>a_{k_j}. So one can find an increasing subsequence.

So in either case, there is a monotonic subsequence.

QED, as they don't say.