I would like to verify that my proof on the following statement is correct:
Any regulated function on $[a, b]$ is bounded.
Let $f \in R[a, b]$ (the space of regulated functions on $[a, b]$) be any regulated function.
Then, by definition, $\forall \varepsilon > 0: \exists \phi \in S[a, b]$ (the space of step functions on $[a, b]$) such that $\|\phi - f\|_\infty < \varepsilon$. Note that all step functions are bounded, i.e. $\phi [a, b]$ is a bounded function.
Now, by definition, $\|\phi - f\|_\infty = \sup_{x \in [a, b]}|\phi - f| < \varepsilon$.
Since the supremum of a function can only be bounded if the function is bounded, it follows that $|\phi - f|$ is a bounded function. But $\phi$ is bounded, so $f$ must also be a bounded function.
i.e. any regulated function on $[a, b]$ is bounded.