Let $(X,A)$ be a measurable space and $f:(X,A) \rightarrow (\mathbb{R}, B(\mathbb{R}))$ be a non-negative bounded function.
Prove there exists a sequence of non-negative simple functions $(f_{n})_{n \in \mathbb{N}} $ so that $\lim_{n \to \infty}\Big(\sup_{x \in X}\big|f_{n}(x) - f(x)\big|\Big) = 0$.
Attempt:
We know that for a non-negative function $f:(X, A) \rightarrow (\mathbb{R}, B(\mathbb{R}))$ we can find a monotone sequence, i.e. $f_{n}(x) \leq f_{n+1}(x)$ for all $x \in X$ so that for all $x \in X$ $$\lim_{n \to \infty}f_{n}(x) = f(x)$$
For example via the typewriter sequence.
This next part is where I am a bit unsure:
Since this sequence of functions is increasing its limit is the supremum. Therefore for some $n \geq N$ $\sup_{x \in X}\Big|f_{n}(x)-f(x)\Big|\Big) = 0$. Hence by taking the limit the result follows.