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 k
_{1}<k_{2}<k_{3}<...

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 k_{1}=b+1. k_{1} isn't a maximal index, so there exists
k_{2}>k_{1} 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.