= Solution
<Fatou lemma> states that for any sequence $(f_n)$ of nonnegative measurable functions on a <measure space>,
$$
\boxed{\int\liminf_{n\to\infty}f_n\,d\mu
\leq
\liminf_{n\to\infty}\int f_n\,d\mu}.
$$
For the proof, define
$$
g_n=\inf_{k\geq n}f_k.
$$
Then $(g_n)$ is a nonnegative increasing sequence and
$$
g_n\uparrow\liminf_{k\to\infty}f_k.
$$
The <monotone convergence theorem> therefore gives
$$
\int\liminf_kf_k\,d\mu
=\lim_{n\to\infty}\int g_n\,d\mu.
$$
For each $k\geq n$, we have $g_n\leq f_k$, so <monotonicity of the Lebesgue integral> yields
$$
\int g_n\,d\mu
\leq\inf_{k\geq n}\int f_k\,d\mu.
$$
Taking the limit in $n$ proves the asserted inequality and the <proof of Fatou lemma>.
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