My instructor briefly discussed a result in lecture that I need help with. Here was the set up.

We assumed that $\{{f}_{n}\}$ is an orthonormal system in ${L}^{2}(0,1)$ and we supposed that there exists an $M>0$ such that $|{f}_{n}(x)|\le M$ a.e for all $n\in \mathbb{N}$. Furthermore, we let $\{{c}_{n}\}$ be a sequence of real numbers such that $\sum _{n=1}^{\mathrm{\infty}}{c}_{n}{f}_{n}$cnfn converges a.e.

They said that it was "clear" that

$\underset{n\to \mathrm{\infty}}{lim}{c}_{n}=0.$

This is not clear or intuitive for me.

After reading through the page, I am still a bit confused. Can someone put together a working proof to clear things up?

We assumed that $\{{f}_{n}\}$ is an orthonormal system in ${L}^{2}(0,1)$ and we supposed that there exists an $M>0$ such that $|{f}_{n}(x)|\le M$ a.e for all $n\in \mathbb{N}$. Furthermore, we let $\{{c}_{n}\}$ be a sequence of real numbers such that $\sum _{n=1}^{\mathrm{\infty}}{c}_{n}{f}_{n}$cnfn converges a.e.

They said that it was "clear" that

$\underset{n\to \mathrm{\infty}}{lim}{c}_{n}=0.$

This is not clear or intuitive for me.

After reading through the page, I am still a bit confused. Can someone put together a working proof to clear things up?