If $f_n \rightarrow f$ a.e. and $\sup_n \|f_n\|_p < \infty$, then $f \in L^p$

Question

I'd like to prove this theorem: Given a sequence $f_n$ in $L_p$ ($1 \le p \le \infty$). If $f_n \rightarrow f$ a.e. and $\sup_n \|f_n\|_p < \infty$, then $f \in L^p$ and $\|f\|_p \le \liminf_{ n \rightarrow \infty} \|f_n\|_p$.

Using Fatou's Lemma I managed to prove the inequality for $1 \le p < \infty$, but I don't know what to do to prove it for $p=\infty$. I think I should use the fact that $\sup_n \|f_n\|_p < \infty$, but I'm stucked.

I also tried to prove that $f \in L^p$ and I know this should be the easiest part to prove, but I'm not sure if what I did is correct. This is my attempt: $\int |f|^p = \int \lim |f_n|^p = \lim \int |f_n|^p < \infty $ ( and again I don't know how to prove this for $p=\infty$).

Answer 1

For the case $p=\infty$, notice that the assumption reads that there exists a constant $C$ such that for almost every $x$ and each $n\geqslant 1$, $\lvert f_n(x)\rvert\leqslant C$ hence $\lvert f(x)\rvert\leqslant C$. Moreover, for almost every $x$, $\lvert f(x)\rvert=\liminf_{n\to\infty}\lvert f_n(x)\rvert \leqslant \liminf_{n\to\infty}\lVert f_n \rVert_\infty$.