I think these three proofs of increasing/decreasing are already very convincing. What I don't understand is the last sentence of the text
"It's not a sure thing that these infinitesimals have anything to do with the non-infinitesimals behaviour of the function"
What will the infinitesimals do with the proof?
I've found the introduction of Infinitesimals on Wikipedia, but it tells nothing related to the mean value theorem. I'm a beginner in calculus and have little knowledge about infinitesimals. Can anyone tell me more about it?