Introductory Real Analysis

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1. Proofs, Sets, and Functions Proofs Sets Functions Mathematical Induction 2. The Structure of R Algebraic and Other Properties of R The Completeness Axiom The Rational Numbers Are Dense in R Cardinality 3. Sequences Convergence Limit Theorems Subsequences Monotone Sequences Bolzano-Weierstrass Theorems Cauchy Sequences Limits at Infinity Limit Superior and Limit Inferior 4. Continuity Continuous Functions Continuity and Sequences Limits of Functions Consequences of Continuity Uniform Continuity Discontinuities and Monotone Functions 5. Differentiation The Derivative Mean Value Theorems Taylor's Theorem L'Hopital's Rule 6. Riemann Integration Existence of the Riemann Integral Riemann Sums Properties of the Riemann Integral Families of Riemann Integrable Functions Fundamental Theorem of Calculus Improper Integrals 7. Infinite Series Convergence and Divergence Absolute and Conditional Convergence Regrouping and Rearranging Series Multiplication of Series 8. Sequences and Series of Functions Function Sequences Preservation Theorems Series of Functions Weierstrass Approximation Theorem 9. Power Series Convergence Taylor Series 10. The Riemann-Stieltjes Theorem Monotone Increasing Integrators Families of Intergrable Functions Riemann-Stieltjes Sums Functions of Bounded Variation Integrators of Bounded Variations 11. The Topology of R Open and Closed Sets Neighborhoods and Accumulation Points Compact Sets Connected Sets Continuous Functions Bibliography. Hints and Answers. Index