I am trying to follow a proof that characterizes regulated functions as those functions $f$ for which there exists a sequence of step functions converging uniformly to $f$. An excerpt of the text I'm using is provided below:
Note that "jump continuous" and "regulated" are synonymous and by $f \in \mathcal{S}(I,E)$ the author just means a regulated function defined on the compact interval $I = [\alpha, \beta]$
I follow all of this proof except for the part where the refined partition $Ʒ_1$ is selected. I don't know why the first partition $Ʒ_0$ doesn't work and I don't know what the motivation is for refining it in the first place. Can anyone explain the purpose of the refined partition and why it is needed?