Dual result of Fatou lemma 
If $\{f_n\}\subset L^+$, $f\in L^+$ such that $\{f_n\}$ is dominated by $f$ where $\int f < \infty$ then $$\limsup\int f_n\leq \int \limsup f_n$$

Attempted proof - Consider the sequence $\{f - f_n\}\subset L^+$ then applying Fatou's lemma we have $$\int (\liminf(f-f_n))\leq \liminf \int (f - f_n)$$ We know that $$\int(\liminf (f - f_n)) = \int \lim_{k\rightarrow \infty} \inf_{n\geq k}(f - f_n)$$ Then from the Monotone Convergence theorem, $$\int \lim_{k\rightarrow \infty}\inf_{n\geq k}(f-f_n) = \lim_{k\rightarrow \infty}\int \inf_{n\geq k}(f - f_n)$$ Therefore we have, $$\lim_{k\rightarrow \infty}\int \inf_{n\geq k}(f - f_n) = \lim_{k\rightarrow \infty}\int \inf_{n\geq k} f - \lim_{k\rightarrow \infty}\int \inf_{n\geq k}f_n = \int  f - \lim_{k\rightarrow \infty}\int \sup_{n\geq k}f_n$$
I am not sure if I am on the right track or where to go from here. Just some hints should help me enough.
Attempted proof #2 - Consider the sequence $\{f - f_n\}\subset L^+$ then applying Fatou's lemma we have 
\begin{align*}
&\int (\liminf(f-f_n))\leq \liminf \int (f - f_n)\\
\Leftrightarrow &\int\liminf f - \int \liminf f_n \leq \liminf\int f - \liminf\int f_n\\
\Leftrightarrow &\int f - \int \limsup f_n \leq \int f - \limsup\int f_n
\end{align*}
Rearranging and cancelling out $\int f$ (which is possible since $\int f <\infty$) then we have $$\limsup\int f_n\leq \int \limsup f_n$$

You may try proving and using the following dual to the monotone convergence theorem:

Suppose $\{g_n\}\subseteq L^+$, $g_n$ decreases pointwise to $g$ , and $g_1$ is integrable. Then, $\int g=\lim\int g_n$.

This is Exercise 2.15 in Folland (1999, p.52).
