Let $a_{n}=\sup_{x\in \mathbb R}|f_{n}(x)|$, such that $a_{n}\to 0$ as $n\to\infty$, If $a_{n}$ is a decreasing sequence and $a_{n}\in (0,1), \forall n$, and $\int_{\mathbb R}|f_{n}(x)|^{2}dx\leq A$ for some $A$, for all $n\geq 1$, Is it true that $\lim_{n\to\infty}\int_{\mathbb R}|f_{n}(x)|^{2}dx=0$?
My guess: Since $a_{n}\to 0$, this means that the sequence $|f_{n}(x)|$ converges to 0 uniformly on $\mathbb R$, hence $|f_{n}(x)|^{2}$ also converges to 0 uniformly on $\mathbb R$, this will imply the result somehow!
Edit: The functions $f_{n}(x)$ are continuous on $\mathbb R$.