As ${g}_{n}$ is an increasing sequence of measurable nonnegative functions we can apply the monotone convergence Theorem^{} to obtain

$${\int}_{X}f\mathit{d}\mu =\underset{n\to \mathrm{\infty}}{lim}{\int}_{X}{g}_{n}\mathit{d}\mu .$$ 

On the other hand, being ${g}_{n}\le {f}_{n}$, we conclude by observing

$$\underset{n\to \mathrm{\infty}}{lim}{\int}_{X}{g}_{n}\mathit{d}\mu =\underset{n\to \mathrm{\infty}}{lim\; inf}{\int}_{X}{g}_{n}\mathit{d}\mu \le \underset{n\to \mathrm{\infty}}{lim\; inf}{\int}_{X}{f}_{n}\mathit{d}\mu .$$ 
