Limits, Limits Everywhere: Paperback: David Applebaum
- Oxford University Press

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A quantity can be made smaller and smaller without it ever vanishing. This fact has profound consequences for science, technology, and even the way we think about numbers. In this book, we will explore this idea by moving at an easy pace through an account of elementary real analysis and, in particular, will focus on numbers, sequences, and series.

Almost all textbooks on introductory analysis assume some background in calculus. This book doesn't and, instead, the emphasis is on the application of analysis to number theory. The book is split into two parts. Part 1 follows a standard university course on analysis and each chapter closes with a set of exercises. Here,
numbers, inequalities, convergence of sequences, and infinite series are all covered. Part 2 contains a selection of more unusual topics that aren't usually found in books of this type. It includes proofs of the irrationality of e and π, continued fractions, an introduction to the Riemann zeta function, Cantor's theory of the infinite, and Dedekind cuts. There is also a survey of what analysis can do for the calculus and a brief history of the subject.

A lot of material found in a standard university course on "real analysis" is covered and most of the mathematics is written in standard theorem-proof style. However, more details are given than is usually the case to help readers who find this style daunting. Both set theory and proof by induction are avoided in
the interests of making the book accessible to a wider readership, but both of these topics are the subjects of appendices for those who are interested in them. And unlike most university texts at this level, topics that have featured in popular science books, such as the Riemann hypothesis, are introduced here. As a result, this book occupies a unique position between a popular mathematics book and a first year college or university text, and offers a relaxed introduction to a fascinating and important branch of mathematics.

Readership: Early undergraduates, those at the end of their secondary education seeking an accessible introduction to university mathematics, and general readers with some background in
mathematics.

David Applebaum, University of Sheffield

David Applebaum obtained his PhD at the University of Nottingham in 1984. After postdoctoral appointments in Rome and Nottingham, he became a lecturer in mathematics at Nottingham Trent University (then Trent Polytechnic) in 1987 and was promoted to reader in 1994 and to a chair in 1998. He was Head of Department 1998-2001. He left Nottingham Trent for a chair in Sheffield in 2004 and served as Head of Department of Probability and Statistics there from 2007-10.

"This book does not offer an easy ride but its informal and enthusiastic literary style hold ones attention. Perhaps mindful of the content of much current popular mathematical exposition, the author draws many illustrations from number theory." - Geoffrey Burton, LMS Newsletter

"The author is able to mix both styles relating informal language to mathematical language and giving proofs that are deep but easy to read and follow." - Luis Sanchez-Gonzalez, the European Mathematical Society

"Written in a style that is easy to read and follow, the author gives clear and succinct explanations and meets his desire for this to be between a textbook and a popular book on mathematics." - John Sykes,
Mathematics in Schools

"Recommended in the Times Higher Education's Textbook Guide 2012." - Noel-Ann Bradshaw, Times Higher Education

"This is an excellent book which should appeal to teachers and pre-University or undergraduate students looking for a hands-on introduction to mathematical analysis." - Mario Cortina Borja, Significance

"The book is devoted to the discussion of one of the most difficult concepts of mathematical analysis, the concept of limits. The presentation is instructive and informal. It allows the author to go much deeper than is usually possible in a standard course of calculus. Moreover, each portion of the material is supplied by an explanation why and what for it is necessary to study (and to
teach) the corresponding part of calculus ... the book can be recommended for interested students as well as for teachers in mathematics." - Zentralblatt MATH

Introduction I Approaching Limits
1: A Whole Lot of Numbers
2: Let's Get Real
3: The Joy of Inequality
4: Where Do You Go To, My Lovely
5: Bounds for Glory
6: You Cannot be Series II Exploring Limits
7: Wonderful Numbers
8: Infinite Products
9: Continued Fractions
10: How Infinite Can You Get?
11: Constructing the Real Numbers
12: Where to Next in Analysis? The Calculus
13: Some Brief Remarks About the History of Analysis Further Reading Apendices
1: The Binomial Theorem
2: The Language of Set Theory
3: Proof by Mathematical Induction
4: The Algebra of Numbers Hints and Selected Solutions

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