I was reading through Royden, Bartle and other Measure Theory textbooks and I saw this problem more than once. If $f \in L(X,\mathcal{A},\mu)$ and $\varepsilon > 0$, then there exists a simple function $\varphi$ such that
$\int |f-\varphi| \; d\mu < \varepsilon.$
I keep having trouble setting up an inequality that can satisfy this so any help is appreciated.