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 $|f_n(x)| \leq M$ a.e for all $n \in \mathbb{N}$. Furthermore, we let $ \{ c_n \}$ be a sequence of real numbers such that $\sum_{n = 1}^{\infty} c_n f_n$ converges a.e.
They said that it was "clear" that $$\lim_{n \to \infty} c_n = 0.$$
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?
Suppose the sum of an infinite series is $\;S\;$ ,and we denote by $\;S_n\;$ the $\;n\,-$th element of the sequence of partial sums of the series. By definition, the series converges iff the limit of $\;S_n\;$ exists finitely ( and in this case $\;S_n\to S\;$) , so:
$$S=\sum_{n=1}^\infty a_n\;,\;\;S_n:=\sum_{k=1}^n a_k\implies a_n=S_n-S_{n-1}\xrightarrow[n\to\infty]{}\stackrel{\text{(arith. of limits)}}S-S=0$$
and we have the basic necessary condition for convergence of infinite series.
In your case you have $\;|c_nf_n|\xrightarrow[n\to\infty]{}0\;$ (no matter whether real of complex numbers), but $\;\{f_n\}\;$ is a bounded sequence...! Finish the argument.