ELEMENTARY REAL ANALYSIS
CONTENTS PROPERTIES OF THE REAL NUMBERS INTRODUCTION THE REAL NUMBER SYSTEM ALGEBRAIC STRUCTURE ORDER STRUCTURE BOUNDS SUPS AND INFS THE ARCHIMEDEAN PROPERTY INDUCTIVE PROPERTY OF IN THE RATIONAL NUMBERS ARE DENSE 1 10 THE METRIC STRUCTURE OF R 1 11 CHALLENGING PROBLEMS FOR CHAPTER SEQUENCES SEQUENCES SEQUENCE EXAMPLES COUNTABLE SETS CONVERGENCE DIVERGENCE BOUNDEDNESS PROPERTIES OF LIMITS ALGEBRA OF LIMITS ORDER PROPERTIES OF LIMITS MONOTONE CONVERGENCE CRITERION 2 10 EXAMPLES OF LIMITS 2 11 SUBSEQUENCES 2 12 CAUCHY CONVERGENCE CRITERION III 2 13 UPPER AND LOWER LIMITS 2 14 CHALLENGING PROBLEMS FOR CHAPTER INFINITE SUMS FINITE SUMS IN?NITE UNORDERED SUMS CAUCHY CRITERION ORDERED SUMS: SERIES PROPERTIES SPECIAL SERIES CRITERIA FOR CONVERGENCE BOUNDEDNESS CRITERION ABSOLUTE CONVERGENCE TESTS FOR CONVERGENCE TRIVIAL TEST DIRECT COMPARISON TESTS LIMIT COMPARISON TESTS RATIO COMPARISON TEST D’ALEMBERT’S RATIO TEST CAUCHY’S ROOT TEST CAUCHY’S CONDENSATION TEST INTEGRAL TEST KUMMER’S TESTS 3 6 10 RAABE’S RATIO TEST 3 6 11 GAUSS’S RATIO TEST 3 6 12 ALTERNATING SERIES TEST 3 6 13 DIRICHLET’S TEST 3 6 14 ABEL’S TEST REARRANGEMENTS UNCONDITIONAL CONVERGENCE CONDITIONAL CONVERGENCE COMPARISON OF ? I=1 AI AND I?IN AI PRODUCTS OF SERIES PRODUCTS OF ABSOLUTELY CONVERGENT SERIES PRODUCTS OF NONABSOLUTELY CONVERGENT SERIES SUMMABILITY METHODS CES`ARO’S METHOD ABEL’S METHOD 3 10 MORE ON IN?NITE SUMS 3 11 IN?NITE PRODUCTS 3 12 CHALLENGING PROBLEMS FOR CHAPTER SETS OF REAL NUMBERS POINTS INTERIOR POINTS ISOLATED POINTS POINTS OF ACCUMULATION BOUNDARY POINTS SETS CLOSED SETS OPEN SETS ELEMENTARY TOPOLOGY COMPACTNESS ARGUMENTS BOLZANO-WEIERSTRASS PROPERTY CANTOR’S INTERSECTION PROPERTY COUSIN’S PROPERTY HEINE-BOREL PROPERTY COMPACT SETS CHALLENGING PROBLEMS FOR CHAPTER CONTINUOUS FUNCTIONS INTRODUCTION TO LIMITS LIMITS (?-? DE?NITION) LIMITS (SEQUENTIAL DE?NITION) LIMITS (MAPPING DE?NITION) ONE-SIDED LIMITS IN?NITE LIMITS PROPERTIES OF LIMITS UNIQUENESS OF LIMITS BOUNDEDNESS OF LIMITS ORDER PROPERTIES COMPOSITION OF FUNCTIONS EXAMPLES LIMITS SUPERIOR AND INFERIOR CONTINUITY HOW TO DE?NE CONTINUITY CONTINUITY AT A POINT CONTINUITY AT AN ARBITRARY POINT CONTINUITY ON A SET PROPERTIES OF CONTINUOUS FUNCTIONS UNIFORM CONTINUITY EXTREMAL PROPERTIES DARBOUX PROPERTY POINTS OF DISCONTINUITY TYPES OF DISCONTINUITY MONOTONIC FUNCTIONS HOW MANY POINTS OF DISCONTINUITY? 5 10 CHALLENGING PROBLEMS FOR CHAPTER MORE ON CONTINUOUS FUNCTIONS AND SETS DENSE SETS NOWHERE DENSE SETS THE BAIRE CATEGORY THEOREM A TWO-PLAYER GAME UNIFORM BOUNDEDNESS CANTOR SETS CONSTRUCTION OF THE CANTOR TERNARY SET AN ARITHMETIC CONSTRUCTION OF K THE CANTOR FUNCTION BOREL SETS SETS OF TYPE G? SETS OF TYPE F? OSCILLATION AND CONTINUITY OSCILLATION OF A FUNCTION THE SET OF CONTINUITY POINTS SETS OF MEASURE ZERO DIFFERENTIATION THE DERIVATIVE DE?NITION OF THE DERIVATIVE DI?ERENTIABILITY AND CONTINUITY THE DERIVATIVE AS A MAGNI?CATION COMPUTATIONS OF DERIVATIVES ALGEBRAIC RULES THE CHAIN RULE INVERSE FUNCTIONS THE POWER RULE CONTINUITY OF THE DERIVATIVE? LOCAL EXTREMA MEAN VALUE THEOREM ROLLE’S THEOREM CAUCHY’S MEAN VALUE THEOREM MONOTONICITY DINI DERIVATES THE DARBOUX PROPERTY OF THE DERIVATIVE 7 10 CONVEXITY 7 11 L’HˆOPITAL’S RULE 7 11 1 L’HˆOPITAL’S RULE 0 FORM 7 11 2 L’HˆOPITAL’S RULE AS X ?? 7 11 3 L’HˆOPITAL’S RULE: ? ?FORM 7 12 TAYLOR POLYNOMIALS 7 13 CHALLENGING PROBLEMS FOR CHAPTER THE INTEGRAL CAUCHY’S FIRST METHOD SCOPE OF CAUCHY’S FIRST METHOD PROPERTIES OF THE INTEGRAL CAUCHY’S SECOND METHOD CAUCHY’S SECOND METHOD (CONTINUED) THE RIEMANN INTEGRAL SOME EXAMPLES RIEMANN’S CRITERIA LEBESGUE’S CRITERION WHAT FUNCTIONS ARE RIEMANN INTEGRABLE? PROPERTIES OF THE RIEMANN INTEGRAL THE IMPROPER RIEMANN INTEGRAL MORE ON THE FUNDAMENTAL THEOREM OF CALCULUS 8 10 CHALLENGING PROBLEMS FOR CHAPTER SEQUENCES AND SERIES OF FUNCTIONS POINTWISE LIMITS UNIFORM LIMITS THE CAUCHY CRITERION WEIERSTRASS M-TEST ABEL’S TEST FOR UNIFORM CONVERGENCE UNIFORM CONVERGENCE AND CONTINUITY DINI’S THEOREM UNIFORM CONVERGENCE AND THE INTEGRAL SEQUENCES OF CONTINUOUS FUNCTIONS SEQUENCES OF RIEMANN INTEGRABLE FUNCTIONS SEQUENCES OF IMPROPER INTEGRALS UNIFORM CONVERGENCE AND DERIVATIVES LIMITS OF DISCONTINUOUS DERIVATIVES POMPEIU’S FUNCTION CONTINUITY AND POINTWISE LIMITS 10 POWER SERIES 10 1 INTRODUCTION 10 2 POWER SERIES: CONVERGENCE 10 3 UNIFORM CONVERGENCE 10 4 FUNCTIONS REPRESENTED BY POWER SERIES 10 4 1 CONTINUITY OF POWER SERIES 10 4 2 INTEGRATION OF POWER SERIES 10 4 3 DI?ERENTIATION OF POWER SERIES 10 4 4 POWER SERIES REPRESENTATIONS 10 5 THE TAYLOR SERIES 10 5 1 REPRESENTING A FUNCTION BY A TAYLOR SERIES 10 5 2 ANALYTIC FUNCTIONS 10 6 PRODUCTS OF POWER SERIES 10 6 1 QUOTIENTS OF POWER SERIES 10 7 COMPOSITION OF POWER SERIES 10 8 TRIGONOMETRIC SERIES 10 8 1 UNIFORM CONVERGENCE OF TRIGONOMETRIC SERIES 10 8 2 FOURIER SERIES 10 8 3 CONVERGENCE OF FOURIER SERIES 10 8 4 WEIERSTRASS APPROXIMATION THEOREM 11 THE EUCLIDEAN SPACES 11 1 THE ALGEBRAIC STRUCTURE OF RN 11 2 THE METRIC STRUCTURE OF RN 11 3 ELEMENTARY TOPOLOGY OF RN 11 4 SEQUENCES IN RN 11 5 FUNCTIONS AND MAPPINGS 11 5 1 FUNCTIONS FROM RN ?R 11 5 2 FUNCTIONS FROM RN ?RM 11 6 LIMITS OF FUNCTIONS FROM RN ?RM 11 6 1 DE?NITION 11 6 2 COORDINATE-WISE CONVERGENCE 11 6 3 ALGEBRAIC PROPERTIES 11 7 CONTINUITY OF FUNCTIONS FROM RN TO RM 11 8 COMPACT SETS IN RN 11 9 CONTINUOUS FUNCTIONS ON COMPACT SETS 11 10 ADDITIONAL REMARKS 12 DIFFERENTIATION ON EUCLIDEAN SPACES 12 1 INTRODUCTION 12 2 PARTIAL AND DIRECTIONAL DERIVATIVES 12 2 1 PARTIAL DERIVATIVES 12 2 2 DIRECTIONAL DERIVATIVES 12 2 3 CROSS PARTIALS 12 3 INTEGRALS DEPENDING ON A PARAMETER 12 4 DI?ERENTIABLE FUNCTIONS 12 4 1 APPROXIMATION BY LINEAR FUNCTIONS 12 4 2 DE?NITION OF DI?ERENTIABILITY 12 4 3 DI?ERENTIABILITY AND CONTINUITY 12 4 4 DIRECTIONAL DERIVATIVES 12 4 5 AN EXAMPLE SU?CIENT CONDITIONS FOR DI?ERENTIABILITY 12 4 7 THE DI?ERENTIAL 12 5 CHAIN RULES 12 5 1 PRELIMINARY DISCUSSION 12 5 2 INFORMAL PROOF OF A CHAIN RULE 12 5 3 NOTATION OF CHAIN RULES 12 5 4 PROOFS OF CHAIN RULES (I) 12 5 5 MEAN VALUE THEOREM 12 5 6 PROOFS OF CHAIN RULES (II) 12 5 7 HIGHER DERIVATIVES 12 6 IMPLICIT FUNCTION THEOREMS 12 6 1 ONE-VARIABLE CASE 12 6 2 SEVERAL-VARIABLE CASE 12 6 3 SIMULTANEOUS EQUATIONS 12 6 4 INVERSE FUNCTION THEOREM 12 7 FUNCTIONS FROM R ?RM 12 8 FUNCTIONS FROM RN ?RM 12 8 1 REVIEW OF DI?ERENTIALS AND DERIVATIVES 12 8 2 DE?NITION OF THE DERIVATIVE 12 8 3 JACOBIANS 12 8 4 CHAIN RULES 12 8 5 PROOF OF CHAIN RULE 13 METRIC SPACES 13 1 INTRODUCTION 13 2 METRIC SPACES—SPECI?C EXAMPLES 13 3 ADDITIONAL EXAMPLES 13 3 1 SEQUENCE SPACES 13 3 2 FUNCTION SPACES 13 4 CONVERGENCE 13 5 SETS IN A METRIC SPACE 13 6 FUNCTIONS 13 6 1 CONTINUITY 13 6 2 HOMEOMORPHISMS 13 6 3 ISOMETRIES 13 7 SEPARABLE SPACES 13 8 COMPLETE SPACES 13 8 1 COMPLETENESS PROOFS 13 8 2 SUBSPACES OF A COMPLETE SPACE 13 8 3 CANTOR INTERSECTION PROPERTY 13 8 4 COMPLETION OF A METRIC SPACE 13 9 CONTRACTION MAPS 13 10 APPLICATIONS OF CONTRACTION MAPS (I) 13 11 APPLICATIONS OF CONTRACTION MAPS (II) 13 11 1 SYSTEMS OF EQUATIONS (EXAMPLE 13 79 REVISITED) 13 11 2 IN?NITE SYSTEMS (EXAMPLE 13 80 REVISITED) 13 11 3 INTEGRAL EQUATIONS (EXAMPLE 13 81 REVISITED) 13 11 4 PICARD’S THEOREM (EXAMPLE 13 82 REVISITED) 13 12 COMPACTNESS 13 12 1 THE BOLZANO-WEIERSTRASS PROPERTY 13 12 2 CONTINUOUS FUNCTIONS ON COMPACT SETS 13 12 3 THE HEINE-BOREL PROPERTY 13 12 4 TOTAL BOUNDEDNESS 13 12 5 COMPACT SETS IN C[A, B] 13 12 6 PEANO’S THEOREM 13 13 BAIRE CATEGORY THEOREM 13 13 1 NOWHERE DENSE SETS 13 13 2 THE BAIRE CATEGORY THEOREM 13 14 APPLICATIONS OF THE BAIRE CATEGORY THEOREM 13 14 1 FUNCTIONS WHOSE GRAPHS “CROSS NO LINES” 13 14 2 NOWHERE MONOTONIC FUNCTIONS 13 14 3 CONTINUOUS NOWHERE DI?ERENTIABLE FUNCTIONS 13 14 4 CANTOR SETS 13 15 CHALLENGING PROBLEMS FOR CHAPTER