1. ## Pointwise convergence

Let $E=[0,\infty)$ and $f_{n}=\frac{1}{n} \chi_{[0,n]}$, n =1,2,.... Show that $\{f_n\}$ converges pointwise to the function $f \equiv 0$ on E, {f_n} is uniformly bounded on E but

$\int\limits_{E} f \ dx \neq \lim_{n \to \infty} \int\limits_{E} f_{n} \ dx$

Does this contradict the bounded convergence theorem.