Question in the proof of Lebesgue's Dominated Convergence Theorem

Question

In the book "Real and Complex analysis 3rd ed." by Walter Rudin,in the proof of Lebesgue's Dominated Convergence Theorem (Chapter 1, page 26), since $\left|f\right|\le g,$ and $f$ is measurable, $f\in L^1(\mu)$. since $\left|f_n-f\right|\le 2g $, then: $$\int_X{2g\,d\mu}\leq\lim_{n\to\infty}\inf\int_X{2g-\left|\,f_n-\,f\right|\,d\mu}$$ but $\left|\,f_n-\,f\right|\ge0$, why could the inequality holds?

Answer 1

Use Fatou lemma and the fact that $2g-|f_{n}-f|\to 2g$ a.e as $f_{n}\to f$ a.e

Answer 2

This is the Fatou Lemma,

https://en.wikipedia.org/wiki/Fatou%27s_lemma

Note that $\liminf_n (2 g - |f_n - f|)=2g$.