Proposition
Let $f$ be a bounded measurable function on $E$. Show that there are $\{\phi_n(x)\}$ and $\{\psi_n(x)\}$ - sequences of simple functions on $E$ such that $\{\phi_n(x)\}$ is increasing and $\{\psi_n(x)\}$ is decreasing and each of these converge to $f$ uniformly on $E$.
By the simple approximation lemma I know that both of the mentioned sequences exists such that $\phi_n(x) \leq f \leq \psi_n(x)$ and $\psi_n(x)-\phi_n(x) < \frac{1}{n}$ for each $n \in \mathbb{N}$.
Using uniform continuity: for some $\varepsilon > 0$ there exists an $n \geq N$ such that $|\phi_n(x)-f| < \frac{1}{n}$. Since simple functions take on a finite number of values, it is reasonable to take a $\max$. So I let $\phi(x)=\max\{\phi_n(x)\}$.
I guess I'm not sure how to pull together the lemma and the definition of simple function.