Let $\mu$ be a measure on $\mathbb{R}^n, \Omega \subseteq \mathbb{R}^n$ be $\mu$-measurable. Let $f:\Omega \rightarrow \mathbb{R}^n$ be $\mu$-measurable, $\forall j \in \mathbb{N}: A_j \subseteq \Omega$ s.t. $f = \sum_{k = 1}^\infty \frac{1}{k}\chi_{A_k}$. i.e. $f_j = \sum_{k = 1}^j \frac{1}{k}\chi_{A_k}$ is an increasing sequence of simple functions convering pointwise to $f$. Show that if $f$ is bounded the convergence is uniform.
My attempt so far: Pick $M \geq 0$ s.t. $\lvert\lvert f \rvert\rvert_\infty \leq M$. Let $N\in \mathbb{N}$ be large enough s.t. $\lvert f_k(x) - f(x)\rvert < \frac{1}{m}$ for $x \in \Omega, m \geq N$
However the final line already uses uniform convergence? I'm not sure how to approach this. Any ideas are appreciated.