**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!!!).