Let $\displaystyle f_n= n \chi_{[\frac{1}{n},\frac{2}{n}]} \,$ on $\displaystyle \mathbb{R}$ for all $\displaystyle n \in \mathbb{N}$ and $\displaystyle f=0 $ on $\displaystyle \mathbb{R}.$

Show that ther does not exist a set of measure zero ,on the complement of which $\displaystyle (f_n)$ is uniformly convergent.