# Real Analysis: A First Course, 2nd edition

• Russell Gordon

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## Overview

 Real Analysis, 2/e is a carefully worded narrative that presents the ideas of elementary real analysis while keeping the perspective of a student in mind. The order and flow of topics has been preserved, but the sections have been reorganized somewhat so that related ideas are grouped together better. A few additional topics have been added; most notably, functions of bounded variation, convex function, numerical methods of integration, and metric spaces. The biggest change is the number of exercises; there are now more than 1600 exercises in the text.

1. Real Numbers.

What Is a Real Number?

Absolute Value, Intervals, and Inequalities.

The Completeness Axiom.

Countable and Uncountable Sets.

Real-Valued Functions.

2. Sequences.

Convergent Sequences.

Monotone Sequences and Cauchy Sequences.

Subsequences.

Supplementary Exercises.

3. Limits and Continuity.

The Limit of a Function.

Continuous Functions.

Intermediate and Extreme Values.

Uniform Continuity.

Monotone Functions.

Supplementary Exercises.

4. Differentiation.

The Derivative of a Function.

The Mean Value Theorem.

Further Topics on Differentiation.

Supplementary Exercises.

5. Integration.

The Riemann Integral.

Conditions for Riemann Integrability.

The Fundamental Theorem of Calculus.

Further Properties of the Integral.

Numerical Integration.

Supplementary Exercises.

6. Infinite Series.

Convergence of Infinite Series.

The Comparison Tests.

Absolute Convergence.

Rearrangements and Products.

Supplementary Exercises.

7. Sequences and Series of Functions.

Pointwise Convergence.

Uniform Convergence.

Uniform Convergence and Inherited Properties.

Power Series.

Taylor's Formula.

Several Miscellaneous Results.

Supplementary Exercises.

8. Point-Set Topology.

Open and Closed Sets.

Compact Sets.

Continuous Functions.

Miscellaneous Results.

Metric Spaces.

Appendix A. Mathematical Logic.

Mathematical Theories.

Statements and Connectives.

Open Statements and Quantifiers.

Conditional Statements and Quantifiers.

Negation of Quantified Statements.

Sample Proofs.

Appendix B. Sets and Functions.

Sets.

Functions.

Appendix C. Mathematical Induction.

Three Equivalent Statements.

The Principle of Mathematical Induction.

The Principle of Strong Induction.

The Well-Ordering Property.