A Concrete Introduction to Real Analysis

This book is an introduction to mathematical analysis. The main feature of the book is a focus on the understanding of basic concepts. It is a book that tries to switch the reader's point of view from “calculus“ to “mathematical analysis“, i.e. from an application of rules to explanations and proofs. The book can be used as an introductory two-semester course. The necessary background is reviewed (including proof by induction and some elementary logic). The topics treated are discrete calculus (proofs by induction, calculus of sums and differences, sums of powers), selected area computations (areas under power function graphs, the computation of π, natural logarithms, the Stirling formula), limits and the Taylor theorem (limits of infinite sequences, series representations, Taylor series), infinite series (positive series, general series, grouping and rearrangement), logic (mathematical philosophy, propositional logic, predicates and quantifiers, proofs), real numbers (field axioms, order axioms, completeness axioms, subsequences and compact intervals, products and fractions), functions (limits and continuity, derivatives), integrals (integrable functions, properties of integrals, numerical computation of integrals, improper integrals, integrals with parameters). Each chapter provides a series of problems for the reader.

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