prove a limit about convergence of norma Given $f_n:\Bbb R \to \Bbb R $ converge to 0 on norma 2. Show that: $$\lim_{n\to\infty}{1\over n}\int_{-n}^n|f_n|dx=0$$ I think it has something to do with C-S inequality but i'm having troubles with it.

Using Daniel Fischer's hint it is enouh to use the C-S inequality: $$\frac{1}{n}\int_{-n}^n|f_n(x)|dx \leq\frac{1}{n}\big(\int_{-n}^n1^2dx\big)^{1/2}\big(\int_{-n}^n|f_n(x)|^2dx\big)^{1/2}$$

$$=\frac{\sqrt{2}}{\sqrt{n}}\|f_n\|_2 \leq \sqrt{2}\|f_n\|_2\rightarrow 0$$.