Equality in the proof of the theorem " every continuous function is integrable"

Question

This is the beginning part of the proof of the theorem that every continuous function is integrable. What is confusing me here is the equality sign in the last expression. Is there a mistake or an explanation?

Answer 1

Note that $f(x)-f(y)<\frac\epsilon{2(b-a)}$ for $x,y\in D$ implies $\sup_D f-\inf_D f\le \frac\epsilon{2(b-a)}$, but in general it does not imply $\sup_D f-\inf_D f< \frac\epsilon{2(b-a)}$.