This question comes from Bartle's "Elements of Integration and Lebesgue Measure" exercise 2L.
Let $f: X\to\mathbb{R}$ be a nonnegative, bounded $\mathcal{X}$-measurable function, where $(X,\mathcal{X})$ is a measurable space. Show that the sequence of functions constructed in lemma 2.11 converges uniformly to $f$.
The sequence mentioned is constructed in the following manner: Fix $n\in\mathbb{N}$ and if $k\in\{0, 1, 2, \dots, n2^n-1\}$ define $E_{k,n} = \{ x\in X : k2^{-n}\le f(x) < (k+1)2^{-n} \} $ and if $k = n2^n$, define $E_{k,n} = \{ x\in X : f(x)\ge n\}$. Those sets are disjoint and their union is $X$. Now we define $\varphi_n(x) = k2^{-n}$ for $x\in E_{k,n}$.
The lemma shows that this sequence is monotone and converges pointwise to $f$. Now if $f$ is bounded, I know that I can take a $N\in\mathbb{N}$ so that $f(x) < N$ for all $x$, so that that last $E_{k,n}$ is empty for all $n>N$. But I don't know what else property of the sequence to use in order to show the convergence. Can someone please help me?