Convergence in the $L^2$ sense but NOT pointwise.

Question

Let

$$f_n(x)=\left\{ \begin{array}{l l} 1 \quad \mbox{ if $\frac12 - \frac1n \leq x \leq \frac12 + \frac1n$,} \\ 0 \quad \mbox{otherwise} \\ \end{array} \right. $$

on the interval $[0,1]$.

How do I show that $f_n \to 0$ in the $L^2$ sense but NOT pointwise.

Answer 1

Hint: $$ \|f_n\|_{L^2}^2= \int_{\frac{1}{2}-\frac{1}{n}}^{\frac{1}{2}+\frac{1}{n}} 1^2 \, dx = \frac{2}{n}. $$

Answer 2

To show that $f_n$ does not converge pointwise to $0$ it is enough to notice that $f_{n}(1/2)=1$ for all $n$.

But you can show that $f_n(x)\to0$ as $n\to \infty$ for all $x\neq 1/2$ where $x\in [0,1]$.