Proof of Fatou lemma
= Proof of Fatou lemma
Set
$$
g_n=\inf_{k\geq n}f_k.
$$
Then $0\leq g_n\uparrow\liminf_k f_k$. The <monotone convergence theorem> and $g_n\leq f_k$ for every $k\geq n$ give
$$
\int\liminf_kf_k\,d\mu
=\lim_n\int g_n\,d\mu
\leq\lim_n\inf_{k\geq n}\int f_k\,d\mu
=\liminf_k\int f_k\,d\mu.
$$