If $f \in L^2(0, \infty)$ then $\lim_{n \to \infty}(n \int_0^n |f(x)|^2dx)^{-1}=0$.

Question

Let $f \in L^2(0, \infty)$. Can you help me to show that $$\lim_{n \to \infty}\left(n \int_0^n |f(x)|^2dx\right)^{-1}=0,$$ please?

Answer 1

This is not true if $f=0$, however if $f\neq 0$ then $\|f\|>0$ and so $$ \lim_{n\to \infty }\frac1{n\|\chi _{(0,n)}f\|^2}=\frac1{\|f\|^2}\lim_{n\to \infty }\frac1n $$