Intermediate Real Analysis

There are a great deal of books on introductory analysis in print today, many written by mathematicians of the first rank. The publication of another such book therefore warrants a defense. I have taught analysis for many years and have used a variety of texts during this time. These books were of excellent quality mathematically but did not satisfy the needs of the students I was teaching. They were written for mathematicians but not for those who were first aspiring to attain that status. The desire to fill this gap gave rise to the writing of this book. This book is intended to serve as a text for an introductory course in analysis. Its readers will most likely be mathematics, science, or engineering majors undertaking the last quarter of their undergraduate education. The aim of a first course in analysis is to provide the student with a sound foundation for analysis, to familiarize him with the kind of careful thinking used in advanced mathematics, and to provide him with tools for further work in it. The typical student we are dealing with has completed a three-semester calculus course and possibly an introductory course in differential equations. He may even have been exposed to a semester or two of modern algebra. All this time his training has most likely been intuitive with heuristics taking the place of proof. This may have been appropriate for that stage of his development.

I Preliminaries.- 1. Sets.- 2. The Set ? of Real Numbers.- 3. Some Inequalities.- 4. Interval Sets, Unions, Intersections, and Differences of Sets.- 5. The Non-negative Integers.- 6. The Integers.- 7. The Rational Numbers.- 8. Boundedness: The Axiom of Completeness.- 9. Archemedean Property.- 10. Euclid’s Theorem and Some of Its Consequences.- 11. Irrational Numbers.- 12. The Noncompleteness of the Rational Number System.- 13. Absolute Value.- II Functions.- 1. Cartesian Product.- 2. Functions.- 3. Sequences of Elements of a Set.- 4. General Sums and Products.- 5. Bernoulli’s and Related Inequalities.- 6. Factorials.- 7. Onto Functions, nth Root of a Positive Real Number.- 8. Polynomials. Certain Irrational Numbers.- 9. One-to-One Functions. Monotonic Functions.- 10. Composites of Functions. One-to-One Correspondences. Inverses of Functions.- 11. Rational Exponents.- 12. Some Inequalities.- III Real Sequences and Their Limits.- 1. Partially and Linearly Ordered Sets.- 2. The Extended Real Number System ?*.- 3. Limit Superior and Limit Inferior of Real Sequences.- 4. Limits of Real Sequences.- 5. The Real Number e.- 6. Criteria for Numbers To Be Limits Superior or Inferior of Real Sequences.- 7. Algebra of Limits: Sums and Differences of Sequences.- 8. Algebra of Limits: Products and Quotients of Sequences.- 9. L’Hôpital’s Theorem for Real Sequences.- 10. Criteria for the Convergence of Real Sequences.- IV Infinite Series of Real Numbers.- 1. Infinite Series of Real Numbers. Convergence and Divergence.- 2. Alternating Series.- 3. Series Whose Terms Are Nonnegative.- 4. Comparison Tests for Series Having Nonnegative Terms.- 5. Ratio and Root Tests.- 6. Kummer’s and Raabe’s Tests.- 7. The Product of Infinite Series.- 8. The Sine and Cosine Functions.- 9. Rearrangements of Infinite Series and Absolute Convergence.- 10. Real Exponents.- V Limits of Functions.- 1. Convex Set of Real Numbers.- 2. Some Real-Valued Functions of a Real Variable.- 3. Neighborhoods of a Point. Accumulation Point of a Set.- 4. Limits of Functions.- 5. One-Sided Limits.- 6. Theorems on Limits of Functions.- 7. Some Special Limits.- 8. P(x) as x ? ± ?, Where P is a Polynomial on ?.- 9. Two Theorems on Limits of Functions. Cauchy Criterion for Functions.- VI Continuous Functions.- 1. Definitions.- 2. One-Sided Continuity. Points of Discontinuity.- 3. Theorems on Local Continuity.- 4. The Intermediate-Value Theorem.- 5. The Natural Logarithm: Logs to Any Base.- 6. Bolzano—Weierstrass Theorem and Some Consequences.- 7. Open Sets in ?.- 8. Functions Continuous on Bounded Closed Sets.- 9. Monotonie Functions. Inverses of Functions.- 10. Inverses of the Hyperbolic Functions.- 11. Uniform Continuity.- VII Derivatives.- 1. The Derivative of a Function.- 2. Continuity and Differentiability. Extended Differentiability.- 3. Evaluating Derivatives. Chain Rule.- 4. Higher-Order Derivatives.- 5. Mean-Value Theorems.- 6. Some Consequences of the Mean-Value Theorems.- 7. Applications of the Mean-Value Theorem. Euler’s Constant.- 8. An Application of Rolle’s Theorem to Legendre Polynomials.- VIII Convex Functions.- 1. Geometric Terminology.- 2. Convexity and Differentiability.- 3. Inflection Points.- 4. Trigonometric Functions.- 5. Some Remarks on Differentiability.- 6. Inverses of Trigonometric Functions. Tschebyscheff Polynomials.- 7. Log Convexity.- IX L’Hôpital’s Rule—Taylor’s Theorem.- 1. Cauchy’s Mean-Value Theorem.- 2. An Application to Means and Sums of Order t.- 3. The O?0 Notation for Functions.- 4. Taylor’s Theorem of Order n.- 5. Taylor and Maclaurin Series.- 6. The Binomial Series.- 7. Tests for Maxima and Minima.- 8. The Gamma Function.- 9. Log-Convexity and the Functional Equation for ?.- X The Complex Numbers. Trigonometric Sums. Infinite Products.- 1. Introduction.- 2. The Complex Number System.- 3. Polar Form of a Complex Number.- 4. The Exponential Function on ?.- 5. nth Roots of a Complex Number. Trigonometric Functions on ?.- 6. Evaluation of Certain Trigonometric Sums.- 7. Convergence and Divergence of Infinite Products.- 8. Absolute Convergence of Infinite Products.- 9. Sine and Cosine as Infinite Products. Wallis’ Product. Stirling’s Formula.- 10. Some Special Limits. Stirling’s Formula.- 11. Evaluation of Certain Constants Associated with the Gamma Function.- XI More on Series: Sequences and Series of Functions.- 1. Introduction.- 2. Cauchy’s Condensation Test.- 3. Gauss’ Test.- 4. Pointwise and Uniform Convergence.- 5. Applications to Power Series.- 6. A Continuous But Nowhere Differentiable Function.- 7. The Weierstrass Approximation Theorem.- 8. Uniform Convergence and Differentiability.- 9. Application to Power Series.- 10. Analyticity in a Neighborhood of x0. Criteria for Real Analyticity.- XII Sequences and Series of Functions II.- 1. Arithmetic Operations with Power Series.- 2. Bernoulli Numbers.- 3. An Application of Bernoulli Numbers.- 4. Infinite Series of Analytic Functions.- 5. Abel’s Summation Formula and Some of Its Consequences.- 6. More Tests for Uniform Convergence.- XIII The Riemann Integral I.- 1. Darboux Integrals.- 2. Order Properties of the Darboux Integral.- 3. Algebraic Properties of the Darboux Integral.- 4. The Riemann Integral.- 5. Primitives.- 6. Fundamental Theorem of the Calculus.- 7. The Substitution Formula for Definite Integrals.- 8. Integration by Parts.- 9. Integration by the Method of Partial Fractions.- XIV The Riemann Integral II.- 1. Uniform Convergence and R-Integrals.- 2. Mean-Value Theorems for Integrals.- 3. Young’s Inequality and Some of Its Applications.- 4. Integral Form of the Remainder in Taylor’s Theorem.- 5. Sets of Measure Zero. The Cantor Set.- XV Improper Integrals. Elliptic Integrals and Functions.- 1. Introduction. Definitions.- 2. Comparison Tests for Convergence of Improper Integrals.- 3. Absolute and Conditional Convergence of Improper Integrals.- 4. Integral Representation of the Gamma Function.- 5. The Beta Function.- 6. Evaluation of ?0+? (sin x)/x dx.- 7. Integral Tests for Convergence of Series.- 8. Jacobian Elliptic Functions.- 9. Addition Formulas.- 10. The Uniqueness of the s, c, and d in Theorem 8.1.- 11. Extending the Definition of the Jacobi Elliptic Functions.- 12. Other Elliptic Functions and Integrals.