I'm working through Royden & Fitzpatrick's Real Analysis, in the beginning of section 6.3 it reads and I quote: "Lebesgue's theorem tells us that a monotone function on an open interval is differentiable almost everywhere. Therefore the difference of two increasing functions on an open interval also is differentiable almost everywhere."
For this to be true, it means that the difference of two increasing functions on an open interval is monotone, but I do not see how that is immediately obvious.
Any clarification is appreciated.