Let $\{f_{n}(x)\}$ be a sequence of continuous positive real valued functions on $\mathbb R$. If $a_{n}=\sup_{x\in \mathbb R}|f_{n}(x)|$, such that $a_{n}\to 0$ as $n\to\infty$, and $a_{n}$ is a decreasing sequence, with $a_{n}\in (0,1), \forall n$, 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!
I asked this before but I got one answer which is not applied for the edit (uniform-convergence-and-integration)