Problem Statement: Let be continuous, , and exists for every . Let be increasing on . Show is increasing on .
Proof: We wish to show that is increasing on which means that for .
Since we know that is continuous and differentiable we may use the Mean Value Theorem. Let , then the MVT implies that there is some such that . Consider . We are given that is increasing on , this implies that for all . Thus, . Since was arbitrary in and for all this implies that for every .
Thus, is increasing on .
Reflection: When I first approached this problem I think I was over-thinking it. I got the equality I have above, but I was worried about concluding that for all since I only showed it was equal to some . The thing that saves us is that our was arbitrary, so we can define for every and show that no matter which we’re looking at . The only reason we’re able to conclude anything in this proof is because we know is increasing. Without that we wouldn’t be able to say anything.