# 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 $X$ lies in the closed bounded region $K_0$. We can divide it into two equally-sized closed bounded regions, $K'_0$ and $K''_0$, 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 $X$; choose one of these and designate it $K_1$. Repeat this construction to get an even smaller closed bounded region containing infinitely many elements of $X$, called $K_2$. By iterating this construction countably infinitely many times, we obtain an infinite sequence of closed bounded regions $K_n$, each of which is half as small as its predecessor and which also and each of which also contains infinitely many elements of $X$.

Finally, we choose elements $x_n \in K_n \cap X \forall n$. By construction, the sequence $x_n$ approaches $\cap_n K_n$, which is simply a point, and we are done.