I have been struggling with the following exercise:
Let $\{f\}_n\subseteq C[a,b]$ be a cauchy sequence w.r.t the $L^2$ norm, s.t. for every $[c,d]\subseteq [a.b]$ we have $\lim_{n\to \infty} \int_c^d f_n(x)dx=0$. Prove that $\int_a^b(f_n(x))^2 dx\to_{n\to \infty}0$.
This exercise isn't intended to be solved with any measure-theoretical tools. In Particular, it is given in a context where $L^2 [a,b]$ is defined to be the Hilbert-completion of $C[a,b]$ w.r.t. $L^2$ norm.
Any help would be appreciated.