>>4
Look at the assumptions of the proof and what the Increment Theorem says. The proof assumes that du/dx, which is the same as u'(x), exists and that Δx is a non-zero infinitesimal. The Increment Theorem says that if u'(x) exists and Δx is infinitesimal, then Δu is infinitesimal (and can be written in a certain form that does not concern us). That's the Δu that the proof uses, and the theorem is mentioned to justify st(Δu)=0. If Δu was not known to be infinitesimal, st(Δu) could be any real number.