Let $f$ be a bounded measurable function on $E$. Show that there are sequences of simple functions on $E$, $\{\varphi_{n}\}$ and $\{\psi_{n}\}$ such that $\{\varphi_{n}\}$ is increasing, $\{\psi_{n}\}$ is decreasing, and each of these sequences converge to $f$ uniformly on $E$.
In order to approach this problem, I was given the hint of using the Simple Approximation Lemma. This Lemma guarantees to us the existence of an increasing sequence $\{\gamma_{n}\}$ that converges pointwise to $f$ on $E$.
Then, and this is the first part of my question: Since $f$ is bounded, we have $|f(x)|\leq M$ $\forall x$. So, the function $f+M$ is both bounded and measurable, and we are given that $\{\gamma_{n}\}$ converges pointwise to $f$, and is increasing. Now, can we then let $\varphi_{n} = \gamma_{n} - M$, where $\{\varphi_{n} \}$ is a sequence that increases pointwise to $f$? And for another sequence $\{\eta_{n}\}$ increasing pointwise to $f$, can we let $\psi_{n} = -\eta_{n} $, and have $\{\psi_{n}\}$ be a sequence decreasing to $f$?
The second part of my question is: once I have my two sequences converging pointwise, how do I show that they converge uniformly? I believe it has something to do with the fact that $f$ is bounded $\forall x$ by a constant.