I want to prove that given a sequence [tex]f_n[/tex] in [tex]L^p[/tex]. If [tex]f_n \rightarrow f[/tex] a.e. and [tex]\sup_n \| f_n\| _ p < \infty[/tex], then [tex]f\in L^p[/tex] and [tex]\|f\|_p \le \liminf_{n\rightarrow \infty} \| f_n \|_p[/tex].

I'm really lost. I'm stuck on this exercise since yesterday, but the only thing I managed to understand is that I think I should use Fatou's lemma to prove the inequality.