I'm struggling with the following problem:
Suppose that $f$ and $f_n$ are analytic and bounded on $\mathbb{R}$, for every $n\in \mathbb N$, and their limit at infinity are zero, i.e.
$$ \lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} f_n(x) = 0.$$
In addition, for every compact $K \subset \mathbb R$ we have uniform convergence
$$\|f - f_n\|_{\infty,K} = \sup_{x\in K}|f(x)-f_n(x)| \rightarrow 0.$$
Is this enough to conclude that we have $L_2$ convergence on $\mathbb R$ ? (i.e. $$ \|f - f_n\|_{L_2}^2 = \int_{0}^{\infty} (f-f_n)^2 dx \rightarrow 0 ~~??)$$
Many thanks!