INSEAD Library

Includes bibliographical references and index

Digitized

Elementary Analysis: the Theory of Calculus Contents Preface 1 Introduction 1 The Set N of Natural Numbers .......................................... 2 The Set Q of Rational Numbers ......................................... 3 The Set R of Real Numbers ............................................... 4 The Completeness Axiom .................................................. 5 The Symbols +oo and --oo ................................................ 6 * A Development of R......................................................... 2 Sequences 7 Limits of Sequences .......................................................... 8 A Discussion about Proofs ................................................ 9 Limit Theorems for Sequences........................................... 10 Monotone Sequences and Cauchy Sequences . ..................... 11 Subsequences................................................................... 12 lim sup's and lim inf's .......................................................... 13 * Some Topological Concepts in Metric Spaces ...................... 14 Series ............................................................................... 15 Alternating Series and Integral Tests .................................... 16 * Decimal Expansions of Real Numbers ................................ v 1 1 6 12 19 27 28 31 31 37 43 54 63 75 79 90 100 105 x Contents 3 Continuity 115 17 Continuous Functions .................................................... 115 18 Properties of Continuous Functions ................................ 126 19 Uniform Continuity ......................................................... 132 20 Limits of Functions ......................................................... 145 21 * More on Metric Spaces: Continuity ......................... 156 22 * More on Metric Spaces: Connectedness ........................ 164 4 Sequences and Series of Functions 171 23 Power Series ................................................................... 171 24 Uniform Convergence ..................................................... 177 25 More on Uniform Convergence ........................................ 184 26 Differentiation and Integration of Power Series . .............. 192 27 * Weierstrass's Approximation Theorem .......................... 200 5 Differentiation 205 28 Basic Properties of the Derivative .................................... 205 29 The Mean Value Theorem ............................................... 213 30 * L'Hospital's Rule ..................................................... 222 31 'Taylor's Theorem ...................................................... 230 6 Integration 243 32 The Riemann Integral ..................................................... 243 33 Properties of the Riemann Integral .................................. 253 34 Fundamental Theorem of Calculus ................................. 261 35 * Riemann-Stieltjes Integrals ..................................... 268 36 * Improper Integrals......................................................... 292 37 * A Discussion of Exponents and Logarithms . ................ 299 Appendix on Set Notation Selected Hints and Answers References Symbols Index Index 309 311 341 345 347