# Lp spaces

### Lp spaces

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

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.
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