A First Course in Analysis

Background Number Systems.- §1. Counting: The Natural Numbers.- §2. Measurement: The Rational Numbers.- The Axioms of Ordered Fields.- §3. Decimal Representation. Irrationals.- I ANALYSIS.- 1 Approximation: The Real Numbers.- §1. Least Upper Bound.- §2. Completeness. Nested Intervals.- §3. Bounded Monotonic Sequences.- §4. Cauchy Sequences.- §5. The Real Number System.- §6. Countability.- Appendix. The Fundamental Theorem of Algebra. Complex Numbers.- 2 The Extreme-Value Problem.- §1. Continuity, Compactness, and the Extreme-Value Theorem.- §2. Continuity of Rational Functions. Limits of Sequences.- Appendix. Completion of the Proof of the Fundamental Theorem of Algebra.- §3. Sequences and Series of Reals. The Number e.- §4. Sets of Reals. Limits of Functions.- 3 Continuous Functions.- §1. Implicit Functions,$$\sqrt[n]{x}$$The Intermediate-Value Theorem.- §2. Inverse Functions. xr for r ? ?.- §3. Continuous Extension. Uniform Continuity. The Exponential and Logarithm.- §4. The Elementary Functions.- §5. Uniformity. The Heine-Borel Theorem.- §6. Uniform Convergence. A Nowhere Differentiable Continuous function.- §7. The Weierstrass Approximation Theorem.- Summary of the Main Properties of Continuous Functions.- Appendix. A Space-Filling Continuous Curve.- II FOUNDATIONS OF CALCULUS.- 4 Differentiation.- §1. Differential and Derivative. Tangent Line.- §2. The Foundations of Differentiation.- §3. Curve Sketching. The Mean-Value Theorem.- §4. Taylor’s Theorem.- §5. Functions Defined Implicitly.- 5 Integration.- §1. Definitions. Darboux Theorem.- §2. Foundations of Integral Calculus. The Fundamental Theorem of Calculus.- §3. The Nature of Integrability. Lebesgue’s Theorem.- §4. Improper Integral.- §5. Arclength. Bounded Variation.- A Word About the Stieltjes Integral and Measure Theory.- 6 Infinite Series.- §1. The Vibrating String.- §2. Convergence: General Considerations.- §3. Convergence: Series of Positive Terms.- §4. Computation with Series.- §5. Power Series.- §6. Fourier Series.