# Limits

## A New Approach to Real Analysis

• Alan F. Beardon
Textbook

Part of the Undergraduate Texts in Mathematics book series (UTM)

1. Front Matter
Pages i-ix
2. ### Foundations

1. Front Matter
Pages 1-1
2. Alan F. Beardon
Pages 3-9
3. Alan F. Beardon
Pages 10-21
3. ### Limits

1. Front Matter
Pages 23-23
2. Alan F. Beardon
Pages 25-49
3. Alan F. Beardon
Pages 50-59
4. Alan F. Beardon
Pages 60-82
5. Alan F. Beardon
Pages 83-96
4. ### Analysis

1. Front Matter
Pages 97-97
2. Alan F. Beardon
Pages 99-115
3. Alan F. Beardon
Pages 116-130
4. Alan F. Beardon
Pages 131-147
5. Alan F. Beardon
Pages 148-169
6. Alan F. Beardon
Pages 170-182
5. Back Matter
Pages 183-190

### Introduction

Broadly speaking, analysis is the study of limiting processes such as sum­ ming infinite series and differentiating and integrating functions, and in any of these processes there are two issues to consider; first, there is the question of whether or not the limit exists, and second, assuming that it does, there is the problem of finding its numerical value. By convention, analysis is the study oflimiting processes in which the issue of existence is raised and tackled in a forthright manner. In fact, the problem of exis­ tence overshadows that of finding the value; for example, while it might be important to know that every polynomial of odd degree has a zero (this is a statement of existence), it is not always necessary to know what this zero is (indeed, if it is irrational, we may never know what its true value is). Despite the fact that this book has much in common with other texts on analysis, its approach to the subject differs widely from any other text known to the author. In other texts, each limiting process is discussed, in detail and at length before the next process. There are several disadvan­ tages in this approach. First, there is the need for a different definition for each concept, even though the student will ultimately realise that these different definitions have much in common.

### Keywords

Finite Mathematica complex number differential equation equality form function functions inequality integration intermediate value theorem mathematical induction real analysis sets theorem

#### Authors and affiliations

• Alan F. Beardon
• 1
1. 1.Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridgeEngland, UK

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4612-0697-2
• Copyright Information Springer-Verlag New York, Inc. 1997
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-1-4612-6872-7
• Online ISBN 978-1-4612-0697-2
• Series Print ISSN 0172-6056
• Buy this book on publisher's site