Royden states that to prove Fatou's lemma it is ncessary and sufficent to show that if $h$ is any bounded measurable function of finite support for which $0\leq h\leq f$ on $E$, then $$ \int _E h\leq \lim \inf \int_E f_n$$
Why did Royden choose to construct such a function instead of working directly with $f$?
By definition, $\int f$ is the supremum of $\int h$ for $0 \leq h \leq f$ bounded measurable of finite support. So at the very end, you just need to take the sup over such $h$ to get $\int f \leq \liminf \int f_n$. He uses such $h$ is order to apply the bounded dominated convergence theorem, which he proves earlier. This allows him to prove Fatou in terms of dominated convergence instead of giving a completely independent proof (and potentially repeating a bunch of work).
You can also let $g_n=inf_{i \geq n} \{f_n\}$ and note that $\int g_n \leq \int f_n$ for each n. Moreover $g_n$ increases monotonically to lim inf $f_n$ . Hence by monotone convergence theorem $\int g_n = \int lim inf f_n$. Since $\int g_n \leq \int f_n$ we have $\int lim inf f_n \leq lim inf \int f_n$. The final step is to note that pointwise convergence implies lim inf $f_n$ = f
