A Concrete Introduction to Real Analysis: (Textbooks in Mathematics)

Most volumes in analysis plunge students into a challenging new mathematical environment, replete with axioms, powerful abstractions, and an overriding emphasis on formal proofs. This can lead even students with a solid mathematical aptitude to often feel bewildered and discouraged by the theoretical treatment. Avoiding unnecessary abstractions to provide an accessible presentation of the material, A Concrete Introduction to Real Analysis supplies the crucial transition from a calculations-focused treatment of mathematics to a proof-centered approach. Drawing from the history of mathematics and practical applications, this volume uses problems emerging from calculus to introduce themes of estimation, approximation, and convergence. The book covers discrete calculus, selected area computations, Taylor's theorem, infinite sequences and series, limits, continuity and differentiability of functions, the Riemann integral, and much more. It contains a large collection of examples and exercises, ranging from simple problems that allow students to check their understanding of the concepts to challenging problems that develop new material. Providing a solid foundation in analysis, A Concrete Introduction to Real Analysis demonstrates that the mathematical treatments described in the text will be valuable both for students planning to study more analysis and for those who are less inclined to take another analysis class.

Table of Contents:
DISCRETE CALCULUS Introduction Proof by Induction A Calculus of Sums and Differences Sums of Powers Problems SELECTED AREA COMPUTATIONS Introduction Areas under Power Function Graphs The Computation of p Natural Logarithms Stirling's Formula Problems LIMITS AND TAYLOR'S THEOREM Introduction Limits of Infinite Sequences Series Representations Taylor Series Problems INFINITE SERIES Introduction Positive Series General Series Grouping and Rearrangement Problems A BIT OF LOGIC Somemathematical Philosophy Propositional Logic Predicates and Quantifiers Proofs Problems REAL NUMBERS Field Axioms Order Axioms Completeness Axioms Subsequences and Compact Intervals Products and Fractions Problems FUNCTIONS Introduction Basics Limits and Continuity Derivatives Problems INTEGRALS Introduction Integrable Functions Properties of Integrals Numerical Computation of Integrals Problems MORE INTEGRALS Introduction Improper Integrals Integrals with Parameters Problems REFERENCES INDEX