# How does this inequality follow? Fatou's Lemma and DCT

I am reading this answer, and I am not sure how we get $$\int g-\int f\leq \int g-\limsup\int f_n.$$ I see that the integral can distribute over the $$-$$ on the LHS, but I am not seeing how the $$\liminf$$ becomes a $$\limsup$$ on the RHS. Why is it that $$\liminf\int(g-f_n)\leq\int g-\limsup\int f_n?$$ I am sure I am just missing something simple here.

• It easily follows from $|f| \leq g$ and $f_n \to f$ a.e. After this use property of liminf – Si Kucing Jun 2 at 23:14
• See math.stackexchange.com/q/334114/148510. $\liminf(x_n) = - \limsup(-x_n)$ – RRL Jun 2 at 23:15