Proving that show that $(f_n)_n$ admits a subsequence that simply converges on $\Bbb R$.

Question

Let $(f_n)_n$ a sequence of increasing functions from $\Bbb R$ to $[-1;1]$

then show that $(f_n)_n$ admits a subsequence that simply converges on $\Bbb R$.

I don't have any clue how to prove this statement. any help?

Answer 1

saulspatz is right in pointing out that this is just Helly's theorem.

Helly's theorem lets you find a subsequence $(f_{n_k})_k$ that converges to some nondecreasing function $f$ except possibly at points at which $f$ is discontinuous. Since $f$ is nondecreasing, it has only countably many such points. Applying a diagonalization argument, we can extract a subsequence of $(f_{n_k})_k$ that converges everywhere.