Real Analysis A First Course

by Gordon, Russell

• ISBN13: 9780201437270
• ISBN10: 0201437279
• Edition: 2nd
• Format: Paperback
• Copyright: 2001-06-01
• Publisher: Pearson

Summary

Real Analysis, 2/eis 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

Real Numbers	1	(48)
What Is a Real Number?	2	(7)
Absolute Value, Intervals, and Inequalities	9	(11)
The Completeness Axiom	20	(9)
Countable and Uncountable Sets	29	(8)
Real-Valued Functions	37	(12)
Sequences	49	(32)
Convergent Sequences	49	(12)
Monotone Sequences and Cauchy Sequences	61	(8)
Subsequences	69	(7)
Supplementary Exercises	76	(5)
Limits and Continuity	81	(48)
The Limit of a Function	82	(11)
Continuous Functions	93	(6)
Intermediate and Extreme Values	99	(11)
Uniform Continuity	110	(5)
Monotone Functions	115	(9)
Supplementary Exercises	124	(5)
Differentiation	129	(34)
The Derivative of a Function	130	(9)
The Mean Value Theorem	139	(8)
Further Topics on Differentiation	147	(9)
Supplementary Exercises	156	(7)
Integration	163	(46)
The Riemann Integral	165	(5)
Conditions for Riemann Integrability	170	(7)
The Fundamental Theorem of Calculus	177	(6)
Further Properties of the Integral	183	(8)
Numerical Integration	191	(10)
Supplementary Exercises	201	(8)
Infinite Series	209	(32)
Convergence of Infinite Series	210	(6)
The Comparison Tests	216	(6)
Absolute Convergence	222	(6)
Rearrangements and Products	228	(7)
Supplementary Exercises	235	(6)
Sequences and Series of Functions	241	(50)
Pointwise Convergence	242	(5)
Uniform Convergence	247	(5)
Uniform Convergence and Inherited Properties	252	(5)
Power Series	257	(9)
Taylor's Formula	266	(7)
Several Miscellaneous Results	273	(11)
Supplemenary Exercises	284	(7)
Point-Set Topology	291	(50)
Open and Closed Sets	292	(8)
Compact Sets	300	(7)
Continuous Functions	307	(9)
Miscellaneous Results	316	(10)
Metric Spaces	326	(15)
A Mathematical Logic	341	(18)
Mathematical Theories	341	(3)
Statements and Connectives	344	(2)
Open Statements and Quantifiers	346	(3)
Conditional Statements and Quantifiers	349	(3)
Negation of Quantified Statements	352	(1)
Sample Proofs	353	(5)
Some Words of Advice	358	(1)
B Sets and Functions	359	(6)
Sets	359	(2)
Functions	361	(4)
C Mathematical Induction	365	(12)
Three Equivalent Statements	365	(2)
The Principle of Mathematical Induction	367	(4)
The Principle of Strong Induction	371	(3)
The Well-Ordering Property	374	(2)
Some Comments on Induction Arguments	376	(1)
Bibliography	377	(2)
Index	379