first of all, i am confused by the definition of . as i see it, this should be:

that is, we want to represent the change in h between the points and (x_{0}plus "a little bit more (plus/minus)"). what you have doesn't look right at ALL, since as , but the idea should be: "h changes very little, if x does".

writing similar definitions for :

therefore:

this is almost what we want, the only "sticking point" is the term , so to finish the proof, one has to show that:

so that one can conclude: