Prove monotone convergence theorem when $int |f_1| dmu < infty$ holds

This question is from 3.A exercise 20 from Axler’s MIRA book.

Suppose $(X,S,mu)$ is a measure space and $f_1, f_2, dots$ is a monotone sequence of $S$-measurable functions. Define $f: X to [-infty,infty]$ by $$f(x) = lim_{ktoinfty}{f_k(x)}.$$ Prove that if $int|f_1|dmu < infty$, then $$lim_{ktoinfty} int{f_k(x)}dmu = int fdmu. $$

The part that I don’t understand how to use the assumption $int |f_1| < infty$. For example if we assumed that $int |f_1| = infty$, then I guess a counter example could be found to show why the equality won’t hold? I don’t think I am thinking correctly about this problem at all and my intuition is all off.