Monthly Archives: May 2011

Spring 2004 #5

Problem Statement: Let be differentiable. Let such that and . If is between and then there exists such that . Proof: Wlog assume . Fix . Define . Then . Note that and since . and so for sufficiently close to , … Continue reading

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Spring 2008 #2

Problem Statement: Let be a twice differentiable real valued function defined on . Suppose that with and . Prove there exists a such that . Proof: Since is twice differentiable on we know that is differentiable on , which implies that … Continue reading

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Spring 2008 #1

Problem Statement: Let be a compact subset of and a continuous function. Show there exists a such that for every . Proof: Since is compact and is continuous it follows that is compact. Bolzano-Weierstrass gives us that is closed and bounded. This means … Continue reading

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February 1998 #1

Problem Statement: Let be an uncountable collection of open sets in . Let . Prove there exists a countable collection of open intervals satisfying (1) and (2) for every positive integer there exists an such that . Proof: Since the rationals are … Continue reading

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September 2000

Problem Statement: Use the open cover definition of compactness to show that any finite union of compact sets is compact. Proof: Let be compact sets in . We wish to show that is a compact set. Let be an open cover of … Continue reading

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May 1999 #5

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 … Continue reading

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May 1990 #4

Problem Statement: Using properties of the Riemann Integral show that if is Riemann Integrable on and then is uniformly continuous on . Proof: We wish to show that for every there exists a such that for every in it follows that . … Continue reading

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