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.)


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.

See also