Is there any example of a sequence of functions such that $\int f_n\to0$ and $\int f_n^2\nrightarrow0$?

Question

Let $(f_n)$ be a sequence in $L^2(a,b)$, where $-\infty<a<b<+\infty$. If $$\lim_{n\to\infty}\int_a^b f_n(x)dx=0,$$ can we conclude that $$\lim_{n\to \infty}\int_a^b f_n(x)^2dx=0?$$

Thanks.

Answer 1

No (to the question in the post). Take $f_n=f$ for all $n$, where $0\ne f\in L_2$ and $\int_a^b f=0$.

Finding such an $f$ should not prove difficult.