# A Primer of Real Analysis

Dan Sloughter, Furman University

Pub Date: 2009

ISBN 13:

Publisher: Independent

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## Conditions of Use

Attribution-NonCommercial-ShareAlike

CC BY-NC-SA

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## Table of Contents

**Preface
Contents
1 Fundamentals**

- 1.1 Sets and relations
- 1.2 Functions
- 1.3 Rational numbers
- 1.4 Real Numbers

**2 Sequences and Series **

- 2.1 Sequences
- 2.2 Infinite series

**3 Cardinality **

- 3.1 Binary representations
- 3.2 Countable and uncountable sets
- 3.3 Power sets

**4 Topology of the Real Line **

- 4.1 Intervals
- 4.2 Open sets
- 4.3 Closed sets
- 4.4 Compact Sets

**5 Limits and Continuity **

- 5.1 Limits
- 5.2 Monotonic functions
- 5.3 Limits to infinity and infinite limits
- 5.4 Continuous Functions

**6 Derivatives **

- 6.1 Best linear approximations
- 6.2 Derivatives
- 6.3 Mean Value Theorem
- 6.4 Discontinuities of derivatives
- 6.5 l’Hˆopital’s rule
- 6.6 Taylor’s Theorem

**7 Integrals **

- 7.1 Upper and lower integrals
- 7.2 Integrals
- 7.3 Integrability conditions
- 7.4 Properties of integrals
- 7.5 The Fundamental Theorem of Calculus
- 7.6 Taylor’s theorem revisited
- 7.7 An improper integral

**8 More Functions**

- 8.1 The arctangent function
- 8.2 The tangent function
- 8.3 The sine and cosine Functions
- 8.4 The logarithm function
- 8.5 The exponential function

**Index **

## About the Book

This is a short introduction to the fundamentals of real analysis. Although the prerequisites are few, I have written the text assuming the reader has the level of mathematical maturity of one who has completed the standard sequence of calculus courses, has had some exposure to the ideas of mathematical proof (including induction), and has an acquaintance with such basic ideas as equivalence relations and the elementary algebraic properties of the integers.

## About the Contributors

### Author(s)

**Dan ****Sloughter** has been teaching Furman students since 1986, and became Professor of Mathematics in 1996. He previously served as an assistant professor at Santa Clara University from 1983-86, and at Boston College from 1981-83. He was also an instructor at Dartmouth College from 1979-81.