May 1999 #5

Problem Statement: Let f be a real valued differentiable function on (0,2) whose derivative f'(x) is bounded on (0,2).  Show that \{f(\dfrac{1}{n})\} converges.

Proof: First note that \dfrac{1}{n}\in (0,2) for every n\in\mathbb{N}. Furthermore we know that f'(x) is bounded so there exists a real number M such that |f'(x)|\leq M for every x\in (0,2). Since f is differentiable on (0,2) we may apply the Mean Value Theorem to any closed interval contained in (0,2).

Let n\in\mathbb{N}. Since [\dfrac{1}{n+1},\dfrac{1}{n}] is contained in (0,2) we know f is differentiable and continuous on the interval. Then by the Mean Value Theorem there exists a c\in (\dfrac{1}{n+1},\dfrac{1}{n}) such that |f'(c)|=|\dfrac{f(\dfrac{1}{n+1})-f(\dfrac{1}{n})}{\dfrac{1}{n+1}-\dfrac{1}{n}}|\leq M. This implies that |f(\dfrac{1}{n})-f(\dfrac{1}{n+1})|\leq M |\dfrac{1}{n}-\dfrac{1}{n+1}|.

Let \varepsilon >0. Then since \{\dfrac{1}{n}\} converges it is also Cauchy. Thus,  we know there exists an N\in\mathbb{N} such that |\dfrac{1}{n}-\dfrac{1}{n+1}|<\dfrac{\varepsilon}{M} for every n\geq N.

This implies that for n\geq N it holds that |f(\dfrac{1}{n})-f(\dfrac{1}{n+1})|\leq M|\dfrac{1}{n}-\dfrac{1}{n+1}|<M\dfrac{\varepsilon}{M}=\varepsilon. Thus, \{f(\dfrac{1}{n})\} is Cauchy and so it converges.

\Box

Reflection:  When I first attacked this problem I tried to use that f is continuous when x=0, 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 (0,\dfrac{1}{n}) 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!!!).

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This entry was posted in Analysis, Cauchy, Differentiable, Math, MVT. Bookmark the permalink.

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