Amazing and Aesthetic Aspects of Analysis
Lively prose and imaginative exercises draw the reader into this unique introductory real analysis textbook. Motivating the fundamental ideas and theorems that underpin real analysis with historical remarks and well-chosen quotes, the author shares his en
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Amazing and Aesthetic Aspects of Analysis
Amazing and Aesthetic Aspects of Analysis
Paul Loya
Amazing and Aesthetic Aspects of Analysis
123
Paul Loya Department of Mathematics Binghamton University Binghamton, NY USA
ISBN 978-1-4939-6793-3 ISBN 978-1-4939-6795-7 https://doi.org/10.1007/978-1-4939-6795-7
(eBook)
Library of Congress Control Number: 2016958470 © Paul Loya 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by the registered company Springer Science+Business Media, LLC part of Springer Nature The registered company address is: 233 Spring Street, New York, NY 10013, U.S.A.
Preface
I have truly enjoyed writing this book on some amazing and aesthetic aspects of analysis. Admittedly, some of the writing is too overdone (e.g., overuse of colloquial language and abundant alliteration at times). But what can I say? I was having fun. The sections of the book with an H are meant to be optional or just for fun and don’t interfere with other sections, besides perhaps other starred sections. Most of the quotations that you’ll find in these pages are taken from the website http://www-gap.dcs.st-and.ac.uk/history/Quotations/. The contents of this book are based on lectures I have given to Binghamton University students taking our fall semester undergraduate real analysis course from 2003 to about 2006. The audience consisted of math majors, including actuarial students, as well as students from the fields of chemistry, computer science, economics, and physics, among others. In order to interest such a diverse body of students, I wanted to write a book that not only teaches the fundamentals of analysis, but also shows its usefulness, beauty, and excitement. I also wanted a book that is personal, in which the students come with me on a journey through some amazing and aesthetic aspects of analysis. There are no derivatives or integrals in this book. This is on purpose, because I wanted to focus on the “eleme
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