I would like to know if this proof of the lemma is correct and full of all the details.
Fatou's Lemma. Let a sequence $\{f_n\}$ of non-negative measurable function. Then $$\int_X \liminf_{n\to\infty} f_n\;d\mu\le\liminf_{n\to\infty}\int_X f_n \;d\mu$$
Proof. The sequence $g_k:X\to [0,\infty]$, $g_k:=\inf_{n\ge k} f_n$ has the following properties:
$(a)\;$$g_k\le g_{k+1}$ for all $k\in \mathbb{N}$, and $g_k$ is measurable for all $k$;
$(b)\;$ $g_k\le f_k$ for all $k\in\mathbb{N}$;
$(c)\;$ $\liminf_{n\to\infty} f_n:=\sup_{k\in\mathbb{N}}\inf_{n\ge k}f_k=\sup_{k\in\mathbb{N}}g_k=\lim_{k\to\infty} g_k.$
Observe that $$g_k\le g_n\le f_n\;\text{for all}\;n\ge k\quad\Rightarrow\quad\int_Xg_k\;d\mu\le \int_X f_n\;d\mu\quad\text{for all}\;n\ge k.$$ Therefore $$ \int_X g_k\;d\mu \le\inf_{n\ge k}\int_X f_n\;d\mu.$$
By the Monotone Convergence Theorem we have that $$\int_X\lim_{k\to\infty} g_k\;d\mu=\lim_{k\to\infty}\int_X g_k\;d\mu$$, then $$\color{BLUE}{\int_X\liminf_{n\to\infty} f_n\;d\mu}=\lim_{k\to\infty}\int_X g_k\;d\mu\color{BLUE}{\le}\lim_{k\to\infty}\inf_{n\ge k}\int_X f_n\;d\mu=\sup_{k\in\mathbb{N}}\inf_{n\ge k}\int_X f_n\;d\mu=\color{BLUE}{\liminf_{n\to\infty}\int_X f_n\;\ d\mu}.$$
How do you know that the $g_n$ are measurable? You simply integrate them without comment. But even if you already have other theorems to rely on for this, you should at least mention that fact before integrating $g_n$.
Other than that, it looks okay.