INTRODUCTION TO REAL ANALYSIS

Transcription

1 INTRODUCTION TO REAL ANALYSIS Michael J. Schramm LeMoyne College PRENTICE HALL Upper Saddle River, New Jersey 07458

2 Contents Preface x PART ONE: PRELIMINARIES Chapter 1: Building Proofs A Quest for Certainty Proofs as Chains Statements Connectives Proof by Contradiction Caution! The Siren Song of Contradiction Disjoined Conclusions Proof by Cases Disjoined Hypotheses Open Statements and Quantifiers Proving Universal Statements Proving Existential Statements Negating a Quantified Statement The Forward-Backward Method A Bigger and Better Example Sets A Glossary Plain as the Nose on Your Face? 31 in

3 IV Chapter 2: Finite, Infinite, and Even Bigger Cardinalities Infinite Sets Uncountable Sets 39 Chapter 3: Algebra of the Real Numbers The Rules of Arithmetic Fields 44 C h a p t e r 4: O r d e r i n g, Intervals, a n d N e i g h b o r h o o d s Orderings The Ordering of the Natural Numbers Well-ordering and Induction Organizing Proofs by Induction Strong Induction Ordered Fields Absolute Value and Distance Intervals When Should We Draw Pictures? Neighborhoods 72 PART TWO: THE STRUCTURE OF THE REAL NUMBER SYSTEM Chapter 5: Upper Bounds and Suprema Upper and Lower Bounds The Least Upper Bound Axiom 84 Chapter 6: Nested Intervals The Integer Part of a Number the Archimedean Property 89

4 6.2 Nests The Fractional Part of a Number Decimal Expansions 95 Chapter 7: Cluster Points Points and Sets One Point of View Another Point of View Cluster Points Derived Sets The Bolzano-Weierstrass Theorem Closing the Loop 110 Chapter 8: Topology of the Real Numbers Open Sets General Topologies Closed Sets The Structure of Open Sets Functions Direct and Inverse Images Continuous Functions Relative Topologies 142 Chapter 9: Sequences An Approximation Problem Convergence Convergent Sequences Sequences and Order Sequences and Algebra Sequences and Topology Subsequences 158

5 VI Chapter 10: Sequences and the Big Theorem Convergence Without Limits Monotone Sequences A Recursively Defined Sequence The Bolzano-Weierstrass Theorem (Revisited) The Converse of Theorem 9.18? Cauchy Sequences Closing the Loop 176 Chapter 11: Compact Sets The Extreme Value Theorem The Covering Property The Heine-Borel Theorem Closing the Loop 194 Chapter 12: Connected Sets The Intermediate Value Theorem Disconnections The Big Theorem Sails into the Sunset Closing the Loop Continuous Functions and Intervals A Comment on Calculus 204 PART THREE: TOPICS FROM CALCULUS Chapter 13: Series We Begin on a Cautious Note Basic Convergence Theorems Series with Positive Terms 210

6 Vll 13.4 Series and the Cauchy Criterion Comparison Tests The Integral Test The Ratio Test Two More Tests for Positive Series Alternating Series Absolute and Conditional Convergence Cauchy Products 228 Chapter 14: Uniform Continuity Uniform Continuity 234 Chapter 15: Sequences and Series of Functions Pointwise Convergence Uniform Convergence Topology of Function Spaces The Weierstrass M-Test Power Series 257 Chapter 16: Differentiation A New Slant on Derivatives Order of Magnitude Estimates Basic Differentiation Theorems The Mean Value Theorem The Meaning of the Mean Value Theorem Taylor Polynomials Taylor Series 277

7 VU1 Chapter 17: Integration Upper and Lower Riemann Integrals Oscillations Integrability of Continuous Functions The Fundamental Theorems Riemann-Stieltjes Integration Riemann-Stieltjes Integrable Functions Integration by Parts 298 Chapter 18: Interchanging Limit Processes A Recurring Problem Double Sequences Integrals and Sequences Derivatives and Sequences Two Applications Integrals with a Parameter "The Moore Theorem" 316 PART FOUR: SELECTED SHORTS Chapter 19: Increasing Functions Discontinuities Discontinuities of Monotone Functions More on Jumps The Cantor Function 324 Chapter 20: Continuous Functions and Differentiability Q.1 Separating the Good from the Bad The Nature of Corners 329

8 IX 20.3 van der Waerden's Function Sets of First and Second Category The Set of Differentiable Functions 337 Chapter 21: Continuous Functions and Integrability Integrable Functions (Revisited) Sets of Content Zero 340 i 21.3 Sets of Measure Zero A Speculative Glimpse at Measure Theory 345 Chapter 22: We Build the Real Numbers Do the Real Numbers Really Exist? Dedekind Cuts The Algebra of Cuts The Ordering of Cuts The Cuts Are the Real Numbers! 356 References and further redding 360 Index 361