Does convergence in $L^2$ imply smallness of $||f_n-f||_{L^2(\mathbb{R}^2\setminus B_R(0)}$ for large $R>0$ independent of $n$?

Question

Let $f_n,f: \mathbb{R}^2\to \mathbb{R}$ be real functions in $L^2$ so that $f_n\to f$ in $L^2(\mathbb{R}^2)$. Does this already imply \begin{align} ||f_n-f||_{L^2(\mathbb{R}^2\setminus B_R(0))}\to 0 \end{align} for all $n$ as $R\to \infty$?