Milne Open Textbooks

Real Analysis


This text is a conventional coverage of Real Analysis for undergraduate students. In it, the real numbers are developed via the Completeness Axiom. The topology of the real numbers is also explored. The coverage culminates in proving the two parts of the Fundamental Theorem of Calculus.

pdf icon Real Analysis


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Introduction: A Quick History of the Development of Real Analysis

1 The Natural Numbers, the Rational Numbers and their Arithmetic

1.1 Exercises

2 Preliminaries Concerning Sets and Functions

2.1 Exercises

3 Moving from Q to R

3.1 Exercises

4 Cardinality

4.1 Exercises

5 Sequences and Series

5.1 Sequences

5.2 Series

Convergence Tests

5.3 Exercises

6 The Topology of the Real Numbers

6.1 Exercises

7 Limits and Continuity

7.1 Exercises

8 The Derivative

8.1 Exercises

9 The Riemann Integral

9.1 Exercises

10 What comes next in Real Analysis?

11 Supplemental Exercises


List of Definitions and Axioms

Alphabetical Index

Gary Towsley

Gary Towsley obtained his Ph.D. in Mathematics in 1975 from the University of Rochester in the field of Compact Riemann Surfaces. He had begun teaching mathematics at SUNY Geneseo in 1974 and continued there until his retirement in June, 2020. He has taught Real Analysis to students at Geneseo almost every other semester. In 2000, he received from  the Mathematical Association of America the Deborah and Franklin Tepper Haimo Award for Distinguished College and University Teaching of Mathematics.