$f_n\to f$ almost everywhere and $\|f_n\|_{L^2}\to \|f\|_{L^2}$ implies $\|f_n-f\|_{L^2}\to 0$

Question

Let $f_n\in L^2(\mathbf{R})$ be a sequence of square integrable functions, $f_n\to f$ almost everywhere, and $\|f_n\|_{L^2}\to \|f\|_{L^2}$. Prove that $\|f_n-f\|_{L^2}\to 0$.

I want to use the dominated convergence theorem but I don't know how to control $\|f_n\|_{L^2}^2$.