Let $\mu$ be a measure on $\mathbb{R}^n, \Omega \subseteq \mathbb{R}^n, \mu$-measurable. Let $f:\Omega \rightarrow [0, \infty], \mu$-measurable. Then there exists for every $k \geq 0: A_k\subseteq \Omega, \mu$-mesurable s.t. $f = \sum_{k=1}^\infty \frac{1}{k} \chi_{A_k}$. Show that if f is bounded the convergence is uniform.
I have no idea how to approach this. I have tried the following, but in my opinion this already assumes uniform convergence:
Pick $M \geq 0$ s.t. $\lvert\lvert f \rvert\rvert _ \infty \leq M$. Let $N \in \mathbb{N}$ large enough s.t. $\forall k \geq N : \lvert f_k(x) - f(x) \rvert \leq \frac{1}{k}$, with $x \in \Omega, f_k$ the partial sums from, the series above.