## lebesgue integration question

Q:
Suppose that ${f_n}_{n=1}^{\infty} \subset L^+$ , $f_n \longrightarrow f$ pointwise and $\int f = lim \int f_n < \infty$. Prove that $\int_E f = lim \int_E f_n$ for all $E \in M$. Prove this need not be true if $\int f = lim \int f_n = \infty$.

Attempt:
I take any set $E \in M$ (where M is the sigma-algebra of Lebesgue Measurable sets). I know that $\int_E f_n = \int f_n \chi_E$. what can i do from here? can i say that $f_n \chi_E \longrightarrow f \chi_E$? help would be appreciated! thanks