Let $\psi_n$, $n=1,2,...$ be a sequence of functions in $L^2([0,2\pi])$ such that $\int_0^{2\pi}\psi_n(t)\psi_m(t)dt$ is equal to $1$ for $n=m$ and vanishes for $n \neq m$. If $A \subset [0,2\pi]$ and $A$ is measurable, prove that $$\lim_{n\to\infty} \int_A \psi_n(x)dx = 0.$$
I'm sorry for my lack of an attempt here, I do normally try to post as much work as I have done, but I honestly have no idea where to start with this one. If someone would give me a hint, I would really appreciate it!