Lebesgue's Theorem...

There is a proof for the above theorem in Riesz/Nagy. The theorem concerns the existence of a finite derivative for the class of monotone functions. It proceeds simply enough by proving a Lemma. The Lemma shows that the function as defined is monotone. However, I'm a little bemused by the contradiction at the end of the proof of the Lemma which ends g(x1) < g(eps) <= g(bk) < g(x1). I can see the first and second half in isolation, but where does the g(eps)<=g(bk) come from?