1st Edition

A Basic Course in Real Analysis

By Ajit Kumar, S. Kumaresan Copyright 2014
    250 Pages 101 B/W Illustrations
    by Chapman & Hall

    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 System
    Algebra 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
    Sequences Defined Recursively

    Continuous Functions
    Definition of Continuity
    Intermediate Value Theorem .
    Extreme Value Theorem
    Monotone Functions
    Uniform Continuity
    Continuous Extensions

    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


    Dr. Ajit Kumar is a faculty member at the Institute of Chemical Technology, Mumbai, India. His main interests are differential geometry, optimization and the use of technology in teaching mathematics. He received his Ph.D. from University of Mumbai. He has initiated a lot of mathematicians into the use of open source mathematics software. Dr. S Kumaresan is currently a professor at University of Hyderabad. His initial training was at Tata Institute of Fundamental Research, Mumbai where he earned his Ph.D. He then served as a professor at University of Mumbai. His main interests are harmonic analysis, differential geometry, analytical problems in geometry, and pedagogy. He has authored five books, ranging from undergraduate level to graduate level.

    "… there are some unique features that put this book aside. … a welcome addition to the library of teachers and student alike."
    Zentralblatt MATH 1308

    "… this book describes the basic results of analysis in an extremely clear, straightforward, and well-motivated way. … if you’re looking for a text on the easy end of the spectrum for a course in real analysis, then this book is certainly worth a serious look …"
    MAA Reviews, October 2014