We have a some measure space $(\Omega,\mathcal(A),\mu)$, and a non-negative measurable function $f:(\Omega,A) \to (R,B(\mathbb R))$ which is also bounded above by some real number.
I want to prove that there is a sequence of simple functions such that $$\lim_{i\to \infty}\Bigl(\sup_{x\in \Omega}|f_i(x)-f(x)|\Bigl)=0$$
Of course, we already know there exists some isotone sequence of simple functions converging pointwise to $f$, but I can't figure out how to use the boundedness to turn this convergence into the stronger version we need. Thanks in advance!
Sketch: let $f : \Omega \to [0, M)$ for some $M \in \mathbb{N}$. For any $n \in \mathbb{N}$, split the codomain into disjoint subintervals $B_1, B_2, \ldots, B_m$ of length $\frac{1}{n}$ (so $m = Mn$), then define $A_i = f^{-1}[B_i]$ and
$$f_n = \sum_{i=1}^m v_i \chi_{A_i}$$
where $v_i$ is any value inside the interval $B_i$.