Show that $\int_0^\infty f du = \lim_{⁡n\to\infty}\int_0^n fdu$

Question

Show that if $f$ is Lebesgue integrable on $[0,\infty)$ then $$\int_0^\infty f du = \lim_{⁡n\to\infty}\int_0^n fdu$$

I tried using Fatou's lemma but I failed to obtain the result. How to go through it?

Answer 1

Hint: what can you say about the functions $f_n:[0,\infty)\to\mathbb{R}$ defined $f_n = f\chi_{[0,n]}$?

Added Note that $\int_0^nfd\mu = \int_{[0,\infty)}f_nd\mu$. It is therefore our goal to show $$\exists\lim_{n\to\infty}\int_{[0,\infty)}f_nd\mu = \int_{[0,\infty)}fd\mu$$ Are you familiar with any tools that might suffice for the task?