A Basic Course in Real Analysis

Based on the authors’ combined 35 years of experience in teaching, A Basic Course in Real Analysis introduces students to the aspects of real analysis in a friendly way. The authors offer insights into the way a typical mathematician works observing patterns, conducting experiments by means of looking at or creating examples, trying to understand the underlying principles, and coming up with guesses or conjectures and then proving them rigorously based on his or her explorations. With more than 100 pictures, the book creates interest in real analysis by encouraging students to think geometrically. Each difficult proof is prefaced by a strategy and explanation of how the strategy is translated into rigorous and precise proofs. The authors then explain the mystery and role of inequalities in analysis to train students to arrive at estimates that will be useful for proofs. They highlight the role of the least upper bound property of real numbers, which underlies all crucial results in real analysis. In addition, the book demonstrates analysis as a qualitative as well as quantitative study of functions, exposing students to arguments that fall under hard analysis. Although there are many books available on this subject, students often find it difficult to learn the essence of analysis on their own or after going through a course on real analysis. Written in a conversational tone, this book explains the hows and whys of real analysis and provides guidance that makes readers think at every stage.

Real Number SystemAlgebra of the Real Number System Upper and Lower Bounds LUB Property and Its Applications Absolute Value and Triangle Inequality Sequences and Their Convergence Sequences and Their Convergence Cauchy Sequences Monotone Sequences Sandwich Lemma Some Important Limits Sequences Diverging to Subsequences Sequences Defined Recursively Continuity Continuous Functions Definition of Continuity Intermediate Value Theorem .Extreme Value Theorem Monotone Functions Limits Uniform Continuity Continuous Extensions Differentiation Differentiability of Functions Mean Value Theorems L'Hospital's Rules Higher-order Derivatives Taylor's Theorem Convex Functions Cauchy's Form of the Remainder Infinite Series Convergence of an Infinite Series Abel's Summation by Parts Rearrangements of an Infinite Series Cauchy Product of Two Infinite Series Riemann Integration Darboux Integrability Properties of the Integral Fundamental Theorems of Calculus Mean Value Theorems for Integrals Integral Form of the Remainder Riemann's Original Definition Sum of an Infinite Series as an Integral Logarithmic and Exponential Functions Improper Riemann Integrals Sequences and Series of Functions Pointwise Convergence Uniform Convergence Consequences of Uniform Convergence Series of Functions Power Series Taylor Series of a Smooth Function Binomial Series Weierstrass Approximation Theorem A Quantifiers B Limit Inferior and Limit Superior C Topics for Student Seminars D Hints for Selected Exercises Bibliography Index