convergence in $L^{2}$

Question

Problem:Let $f_n: [0, 1] → R (n = 1, 2,...)$ Be the functions defined by

$f_{n}(x)=\begin{cases}0&\text{ for }x=0 \\ n^{4/7}\sin {1/x^2}&\text{ for }x\in \left ( 0,1/n \right ] \\ 0&\text{ for }x\in \left ( 1/n,1 \right ) \\ 1&\text{ for } x=1\end{cases}$

In what sense (in the Lebesgue spatial norm $L^{2}[0, 1]$, on average, punctually, almost everywhere, to a degree) does the sequence {fn} converge? Justify your answers. Find the limit function.

The "average" convergence comes from the pointwise-convergence, but uniform and $L^2$... I don't know how to test them, please help me

Thanks for any Hints

Answer 1

What do you need to show for uniform convergence? Maybe you can find a sequence $(x_n)$ such that $f_n(x_n)$ does not converge to 0. You may want to make the $\sin(\frac{1}{x^2})$ constant on this sequence.