# Elementary Analysis: The Theory of Calculus

Springer Science & Business Media, Mar 3, 1980 - Mathematics - 264 pages
Designed for students having no previous experience with rigorous proofs, this text on analysis can be used immediately following standard calculus courses. It is highly recommended for anyone planning to study advanced analysis, e.g., complex variables, differential equations, Fourier analysis, numerical analysis, several variable calculus, and statistics. It is also recommended for future secondary school teachers. A limited number of concepts involving the real line and functions on the real line are studied. Many abstract ideas, such as metric spaces and ordered systems, are avoided. The least upper bound property is taken as an axiom and the order properties of the real line are exploited throughout. A thorough treatment of sequences of numbers is used as a basis for studying standard calculus topics. Optional sections invite students to study such topics as metric spaces and Riemann-Stieltjes integrals.

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### Contents

 1 Introduction 1 2 The Set Q of Rational Numbers 6 3 The Set R of Real Numbers 12 4 The Completeness Axiom 19 5 The Symbols +0o and 0o 27 6 A Development of R 28 7 Limits of Sequences 31 8 A Discussion about Proofs 37
 4 Sequences and Series of Functions 171 24 Uniform Convergence 177 25 More on Uniform Convergence 184 26 Differentiation and Integration of Power Series 192 27 Weierstrasss Approximation Theorem 200 5 Differentiation 205 29 The Mean Value Theorem 213 30 LHospitals Rule 222

 9 Limit Theorems for Sequences 43 10 Monotone Sequences and Cauchy Sequences 54 11 Subsequences 63 12 lim sups and lim infs 75 13 Some Tbpological Concepts in Metric Spaces 79 14 Series 90 15 Alternating Series and Integral Tests 100 16 Decimal Expansions of Real Numbers 105 3 Continuity 115 18 Properties of Continuous Functions 126 19 Uniform Continuity 132 20 Limits of Functions 145 Continuity 156 Connectedness 164
 31 Taylors Theorem 230 6 Integration 243 33 Properties of the Riemann Integral 253 34 Fundamental Theorem of Calculus 261 35 RiemannStieltjes Integrals 268 36 Improper Integrals 292 37 A Discussion of Exponents and Logarithms 299 Appendix on Set Notation 309 Answers 311 References 341 Symbols Index 345 Index 347 Copyright