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Limits

A New Approach to Real Analysis

  • Textbook
  • © 1997

Overview

Authors:
  1. Alan F. Beardon

    1. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, England, UK

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  • Beardon is an excellent writer and teacher (Cambridge)
  • By unifying and simplifying all the various notions of limit, the author has successfully presented a unique and novel approach to the subject matter, which has not previously appeared in book form
  • The author defines what is meant by a limit just once, and all of the subsequent limiting processes will be seen as special cases of this one definition

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

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-ix
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  2. Foundations

    1. Front Matter

      Pages 1-1
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    2. Sets and Functions

      • Alan F. Beardon
      Pages 3-9

    3. Real and Complex Numbers

      • Alan F. Beardon
      Pages 10-21

  3. Limits

    1. Front Matter

      Pages 23-23
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    2. Limits

      • Alan F. Beardon
      Pages 25-49

    3. Bisection Arguments

      • Alan F. Beardon
      Pages 50-59

    4. Infinite Series

      • Alan F. Beardon
      Pages 60-82

    5. Periodic Functions

      • Alan F. Beardon
      Pages 83-96

  4. Analysis

    1. Front Matter

      Pages 97-97
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    2. Sequences

      • Alan F. Beardon
      Pages 99-115

    3. Continuous Functions

      • Alan F. Beardon
      Pages 116-130

    4. Derivatives

      • Alan F. Beardon
      Pages 131-147

    5. Integration

      • Alan F. Beardon
      Pages 148-169

    6. π, γ, e, and n!

      • Alan F. Beardon
      Pages 170-182

  5. Back Matter

    Pages 183-190
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Keywords

About this book

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.

Authors and Affiliations

  • Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, England, UK

    Alan F. Beardon

Bibliographic Information

  • Book Title: Limits

  • Book Subtitle: A New Approach to Real Analysis

  • Authors: Alan F. Beardon

  • Series Title: Undergraduate Texts in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4612-0697-2

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1997

  • Hardcover ISBN: 978-0-387-98274-8Published: 30 October 1997

  • Softcover ISBN: 978-1-4612-6872-7Published: 03 October 2012

  • eBook ISBN: 978-1-4612-0697-2Published: 06 December 2012

  • Series ISSN: 0172-6056

  • Series E-ISSN: 2197-5604

  • Edition Number: 1

  • Number of Pages: IX, 190

  • Topics: Real Functions

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