Let $(f_n)_n \subset L^p, \quad 1\leq p < \infty, \quad f_n \rightarrow f $a.e, $\| f_n \|_p \rightarrow \| f \|_p$ when $n \rightarrow \infty$.
(a) Assume $p=1$ and $g_n = | f_n |+ | f |- | f_n - f |$. Show using fatou's Lemma that $f_n \rightarrow f$ in $L^1$.
(b) For $p>1,$ prove that $f_n \rightarrow f$ in $L^p$ using fatou's Lemma.