Transcription of Problems and Solutions in EAL AND COMPLEX ANALYSIS
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Problems and Solutions in EAL AND COMPLEX ANALYSIS
Problems and Solutions in R EAL AND C OMPLEX A NALYSIS. William J. DeMeo July 9, 2010. c William J. DeMeo. All rights reserved. This document may be copied for personal use. Permission to reproduce this document for other purposes may be obtained by emailing the author at Abstract The pages that follow contain unofficial Solutions to Problems appearing on the comprehensive exams in ANALYSIS given by the Mathematics Department at the University of Hawaii over the period from 1991 to 2007. I have done my best to ensure that the Solutions are clear and correct, and that the level of rigor is at least as high as that expected of students taking the exams. In solving many of these Problems , I benefited enormously from the wisdom and guidance of professors Tom Ramsey and Wayne Smith.
1 REAL ANALYSIS 1 Real Analysis 1.1 1991 November 21 1.(a) Let f nbe a sequence of continuous, real valued functions on [0;1] which converges uniformly to f.Prove that lim n!1f n(x n) = f(1=2) for any sequence fx ngwhich converges to 1=2. (b) Must the conclusion still hold if the convergence is only point-wise?
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