If a series of continuous functions converges pointwise on $[0,1]$, then the partial sums are uniformly bounded on a smaller interval

Question

Let $f_n:[0,1]\rightarrow\mathbb{R}$ be continuous functions, $n\in\mathbb{N}$, such that the series $\sum_{n=1}^\infty f_n(x)$ converges for all $x\in[0,1]$. Prove that for some interval $[a,b]\subseteq[0,1]$ the partial sums of this series are uniformly bounded.

I do not make a solution yet, but I understood that WLOG I can think that $f_n$ is a polynomial (by Stone-Weierstrass theorem). Also it maybe be better to use $[-1,1]$ instead $[0,1]$. Am I right?

Answer 1

Let $s_n(x)$ be the partial sums,which are continuous.

If $x \in [0,1]$ the partial sums at $x$ are bounded,because of the convergence.

Take the sets $A_m=\{x \in [0,1]: \forall n \in \Bbb{N}, |s_n(x)| \leq m\}$

Prove that the sets $A_m$ are closed and that their union is $[0,1]$.

And finally use Baire's category theorem on the complete metric space $[0,1]$ with the usual metric.