This is a theorem proven in my book. As usual, I covered up the proof of the theorem so that I might prove it myself. I came up with essentially the same proof, the only point of contrast being that I tried to explicitly construct a partition, whereas the author just assumed that there exists a partition $\mathcal{P}$ with $\|\mathcal{P} \| < \delta_{\epsilon}$
The simplest partition to choose, as it appears to me, would be one having each pair of adjacent points separated by the same amount. The problem I face is very simple: given $\Delta x = \frac{1}{2} \min \{\delta'_{\epsilon/2},\delta''_{\epsilon/2} \}$, how many points do I need in partition $\mathcal{P}$ to insure $\| \mathcal{P}\| = \Delta x$? I took $a=0$ and $b=2$ and tried various values of $\Delta x$; I conjectured that the number points needed is $\left\lfloor\frac{b-a}{\Delta x} \right\rfloor + 1$, but I am not sure this is correct, nor am I sure of how to prove this, if it is in fact true.