Sequence of Functions Converging to 0 I encountered this question in a textbook. While I understand the intuition behind it I am not sure how to formally prove it.
Define the sequence of functions $(g_n)$ on $[0,1]$ to be $$g_{k,n}(x) = \begin{cases}1 & x \in \left[\dfrac{k}n, \dfrac{k+1}n\right]\\ 0 & \text{ else}\end{cases}$$
where $k \in \{0,1,2,\ldots,n-1\}$.
Prove the following statements:
1) $g_n \to 0$ with respect to the $L^2$ norm.
2) $g_n(x)$ doesn't converge to $0$ at any point in $[0,1]$.
3) There is a subsequence of $(g_n)$ that converges pointwise to $0$.
Thank you for your replies.

1) What is $\lVert g_n\rVert_2$?
2) Show that for any $x$, $g_n(x)=0$ infinitely often and  $g_n(x)=1$ infinitely often.
3) Note that $\frac1n\to 0 $ and $\frac2n\to 0$
