Real Analysis: A First Course / Edition 2by Russell Gordon
Pub. Date: 06/01/2001
|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/i>|
|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.|
Table of Contents
1. Real Numbers.
What Is a Real Number?
Absolute Value, Intervals, and Inequalities.
The Completeness Axiom.
Countable and Uncountable Sets.
Monotone Sequences and Cauchy Sequences.
3. Limits and Continuity.
The Limit of a Function.
Intermediate and Extreme Values.
The Derivative of a Function.
The Mean Value Theorem.
Further Topics on Differentiation.
The Riemann Integral.
Conditions for Riemann Integrability.
The Fundamental Theorem of Calculus.
Further Properties of the Integral.
6. Infinite Series.
Convergence of Infinite Series.
The Comparison Tests.
Rearrangements and Products.
7. Sequences and Series of Functions.
Uniform Convergence and Inherited Properties.
Several Miscellaneous Results.
8. Point-Set Topology.
Open and Closed Sets.
Appendix A. Mathematical Logic.
Statements and Connectives.
Open Statements and Quantifiers.
Conditional Statements and Quantifiers.
Negation of Quantified Statements.
Some Words of Advice.
Appendix B. Sets and Functions.
Appendix C. Mathematical Induction.
Three Equivalent Statements.
The Principle of Mathematical Induction.
The Principle of Strong Induction.
The Well-Ordering Property.
Some Comments on Induction Arguments.
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