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?
2026-04-11 23:46:19.1775951179
Proving that show that $(f_n)_n$ admits a subsequence that simply converges on $\Bbb R$.
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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.