Relation between convergence in $L^2$ and convergence of the integrals

Question

Let $\Omega$ be an open set of the real numbers. If $f_n$ converges to $f$ in $L^2(\Omega)$ that is if $\int_{\Omega} |f_n(x)-f(x)|^2\,dx$ goes to 0 as n goes to infinity, does this imply that $\int_{\Omega} f_n(x)^2\,dx$ goes to $\int_{\Omega} f(x)^2\,dx$ as n goes to infinity ?