# Bolzano-Weierstrass theorem

The Bolzano-Weierstrass theorem is a theorem which states that every infinite set in a closed bounded region must have an infinite convergent subsequence. (In fact, we may even ensure that this infinite convergent subsequence converges arbitrarily quickly, but this is beside the point.)

## Proof

Suppose that the set lies in the closed bounded region . We can divide it into two equally-sized closed bounded regions, and , which can be bounded by smaller neighborhoods. Then by the infinitary pigeonhole principle, at least one of these regions must contain an infinite number of elements of ; choose one of these and designate it . Repeat this construction to get an even smaller closed bounded region containing infinitely many elements of , called . By iterating this construction countably infinitely many times, we obtain an infinite sequence of closed bounded regions , each of which is half as small as its predecessor and which also and each of which also contains infinitely many elements of .

Finally, we choose elements . By construction, the sequence approaches , which is simply a point, and we are done.