Let $(e_n)$ be orthonormal basis of $L^2(a,b)$. Prove that, for every set $A \subset (a,b)$ which has positive Lebesgue measure, $\sum_{n=1}^{\infty} \int_A |e_n(x)|^2 dx = \infty.$ Also, prove that $\sum_{n=1}^{\infty} |e_n(x)|^2 = \infty$ almost everywhere.
I don't know how to prove this. I guess I have to use information that $m(A)>0$ for the first part, but I don't know how.
I know that $||e_n||=1$, so $\int_a^b |e_n(x)|^2dx =1$, but $\int_A |e_n(x)|^2 dx$ is then less or equal 1, so I get nothing from it.
Thanks.