Convergence in $L^2(\Bbb R)$ implies convergence of the norms

Question

If $||f_n-f||_{L^2(\mathbb{R})}\to 0$ is it always true that $||f||_{L^2(\mathbb{R})}=\lim_{n\to\infty}||f_n||_{L^2(\mathbb{R})}$?

Answer 1

Triangle inequality

$$|\, ||f_n ||_2 -|| f ||_2\, |\leq ||f_n-f||_2$$