$f_n$ is continuous on $[a,b]$, and for any $x\in [a,b]$, $\{f_n(x)\}$ is a bounded sequence. Show that for some interval, $f_n$ is uniformly bounded

Question

$f_n$ is continuous on $[a,b]$, and for any $x\in [a,b]$, $\{f_n(x)\}$ is a bounded sequence. Show that for some interval, $f_n$ is uniformly bounded.

What to do? Heine-Borel property? How to find such an interval? For any $x\in [a,b]$, there exists $M_x>0$ such that for all $n$, $|f_n(x)|\leq M_x$. How to bound $M_x$ for $x$ in some nice interval?

Answer 1

$[a,b]=\bigcup_m \bigcap_n\{x: |f_n(x)| \leq m\}$. This is a countable union of closed sets. Apply Baire Category Theorem to get the answer immeditely.

Ref: https://en.wikipedia.org/wiki/Baire_category_theorem