Category Archives: Sequence of Functions

Winter 1993

Problem Statement: Suppose that converges to uniformly and converges to uniformly on where each is continuous on . Prove that converges uniformly to on . Proof: Since is a compact set and are each continuous on it follows that each is bounded … Continue reading

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Winter 1995 #4

Problem Statement: Let be uniformly continuous and let for every . Show converges uniformly. Proof: First I claim that converges pointwise to . Let . Then . Since is continuous and converges to it follows that . This holds for all … Continue reading

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Winter 2004#5

Problem Statement: Let be defined on . (a) Find . (b) State a definition of Riemann Integrable and use it to prove or disprove is Riemann Integrable on . (a) Note that for and that for . We know that if … Continue reading

Posted in Analysis, Math, Riemann Integrable, Sequence of Functions | 2 Comments

Spring 1996 #4

More studying of sequences and series of functions. These boogers are beginning to be a little more manageable. Problem Statement: Consider the sequence of functions for . (a) Prove converges. (b) Is the convergence uniform? Prove your answer. (a) I claim that converges pointwise to … Continue reading

Posted in Analysis, Sequence of Functions, Uniform Convergence | 1 Comment

May 2000#5

Problem Statement: For every and define . Find the function that converges to pointwise. Is the converges uniform? Proof: Fix . Then and so converges to pointwise.  Claim: This convergence is not uniform.  If the convergence is uniform then it follows that for … Continue reading

Posted in Analysis, Math, Sequence of Functions, Uniform Convergence | 1 Comment

Fall 1991 #5

This begins my study of sequences and series of functions. These bad boys have a tendency of tripping me up so I’ve decided to spend some quality time learning their ways. Problem Statement: Let and let be a sequence of real-valued functions on … Continue reading

Posted in Analysis, Math, Sequence of Functions, Uniform Continuity | 2 Comments