Intermediate Real Analysis

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'Hopital'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'Hopital'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.