Problem Statement: Let be a real valued differentiable function on whose derivative is bounded on . Show that converges.
Proof: First note that for every . Furthermore we know that is bounded so there exists a real number such that for every . Since is differentiable on we may apply the Mean Value Theorem to any closed interval contained in .
Let . Since is contained in we know is differentiable and continuous on the interval. Then by the Mean Value Theorem there exists a such that . This implies that .
Let . Then since converges it is also Cauchy. Thus, we know there exists an such that for every .
This implies that for it holds that . Thus, is Cauchy and so it converges.
Reflection: When I first attacked this problem I tried to use that is continuous when , but we are not given that information. After further thought I realized that I was being asking information about the function after having been given information about the derivative, which should be a huge red flag to use the MVT. I then tried to use the interval in my MVT inequality but Jeremy pointed out that we can’t do that since we must have continuity on the closed interval in order to use the MVT. Then I realized I could show convergence by showing Cauchy.
At first glance I didn’t realize all of the little nuances of this problem, but it’s those little nuances that can make or break a qual. I need to keep in mind the hypotheses when I’m taking the qual (in 12 days!!!).