Undergraduate Texts in Mathematics

Background Number Systems.- x1. Counting: The Natural Numbers.- x2. Measurement: The Rational Numbers.- The Axioms of Ordered Fields.- x3. Decimal Representation. Irrationals.- I ANALYSIS.- 1 Approximation: The Real Numbers.- x1. Least Upper Bound.- x2. Completeness. Nested Intervals.- x3. Bounded Monotonic Sequences.- x4. Cauchy Sequences.- x5. The Real Number System.- x6. Countability.- Appendix. The Fundamental Theorem of Algebra. Complex Numbers.- 2 The Extreme-Value Problem.- x1. Continuity, Compactness, and the Extreme-Value Theorem.- x2. Continuity of Rational Functions. Limits of Sequences.- Appendix. Completion of the Proof of the Fundamental Theorem of Algebra.- x3. Sequences and Series of Reals. The Number e.- x4. Sets of Reals. Limits of Functions.- 3 Continuous Functions.- x1. Implicit Functions,$$sqrt[n]{x}$$The Intermediate-Value Theorem.- x2. Inverse Functions. xr for r ? ?.- x3. Continuous Extension. Uniform Continuity. The Exponential and Logarithm.- x4. The Elementary Functions.- x5. Uniformity. The Heine-Borel Theorem.- x6. Uniform Convergence. A Nowhere Differentiable Continuous function.- x7. The Weierstrass Approximation Theorem.- Summary of the Main Properties of Continuous Functions.- Appendix. A Space-Filling Continuous Curve.- II FOUNDATIONS OF CALCULUS.- 4 Differentiation.- x1. Differential and Derivative. Tangent Line.- x2. The Foundations of Differentiation.- x3. Curve Sketching. The Mean-Value Theorem.- x4. Taylor's Theorem.- x5. Functions Defined Implicitly.- 5 Integration.- x1. Definitions. Darboux Theorem.- x2. Foundations of Integral Calculus. The Fundamental Theorem of Calculus.- x3. The Nature of Integrability. Lebesgue's Theorem.- x4. Improper Integral.- x5. Arclength. Bounded Variation.- A Word About the Stieltjes Integral and Measure Theory.- 6 Infinite Series.- x1. The Vibrating String.- x2. Convergence: General Considerations.- x3. Convergence: Series of Positive Terms.- x4. Computation with Series.- x5. Power Series.- x6. Fourier Series.