UNDERSTANDING ANALYSIS

By AMELIA TAYLOR

CONTENTS THE REAL NUMBERS DISCUSSION: THE IRRATIONALITY OF ? 2 SOME PRELIMINARIES THE AXIOM OF COMPLETENESS CONSEQUENCES OF COMPLETENESS CANTOR’S THEOREM EPILOGUE SEQUENCES AND SERIES DISCUSSION: REARRANGEMENTS OF IN?NITE SERIES THE LIMIT OF A SEQUENCE THE ALGEBRAIC AND ORDER LIMIT THEOREMS THE MONOTONE CONVERGENCE THEOREM AND A FIRST LOOK AT IN?NITE SERIES SUBSEQUENCES AND THE BOLZANO–WEIERSTRASS THEOREM THE CAUCHY CRITERION PROPERTIES OF IN?NITE SERIES DOUBLE SUMMATIONS AND PRODUCTS OF IN?NITE SERIES BASIC TOPOLOGY OF R DISCUSSION: THE CANTOR SET OPEN AND CLOSED SETS COMPACT SETS PERFECT SETS AND CONNECTED SETS BAIRE’S THEOREM FUNCTIONAL LIMITS AND CONTINUITY DISCUSSION: EXAMPLES OF DIRICHLET AND THOMAE FUNCTIONAL LIMITS COMBINATIONS OF CONTINUOUS FUNCTIONS CONTINUOUS FUNCTIONS ON COMPACT SETS THE INTERMEDIATE VALUE THEOREM SETS OF DISCONTINUITY THE DERIVATIVE DISCUSSION: ARE DERIVATIVES CONTINUOUS? DERIVATIVES AND THE INTERMEDIATE VALUE PROPERTY THE MEAN VALUE THEOREM A CONTINUOUS NOWHERE-DI?ERENTIABLE FUNCTION SEQUENCES AND SERIES OF FUNCTIONS DISCUSSION: BRANCHING PROCESSES UNIFORM CONVERGENCE OF A SEQUENCE OF FUNCTIONS UNIFORM CONVERGENCE AND DI?ERENTIATION SERIES OF FUNCTIONS POWER SERIES TAYLOR SERIES THE RIEMANN INTEGRAL DISCUSSION: HOW SHOULD INTEGRATION BE DE?NED? THE DE?NITION OF THE RIEMANN INTEGRAL INTEGRATING FUNCTIONS WITH DISCONTINUITIES PROPERTIES OF THE INTEGRAL THE FUNDAMENTAL THEOREM OF CALCULUS LEBESGUE’S CRITERION FOR RIEMANN INTEGRABILITY ADDITIONAL TOPICS THE GENERALIZED RIEMANN INTEGRAL METRIC SPACES AND THE BAIRE CATEGORY THEOREM FOURIER SERIES A CONSTRUCTION OF R FROM Q CHAPTER DISCUSSION: THE IRRATIONALITY OF ? TOWARD THE END OF HIS DISTINGUISHED CAREER, THE RENOWNED BRITISH MATHEMATICIAN GH HARDY ELOQUENTLY LAID OUT A JUSTI?CATION FOR A LIFE OF STUDYING MATHEMATICS IN A MATHEMATICIAN’S APOLOGY, AN ESSAY ?RST PUBLISHED IN 1940 AT THE CENTER OF HARDY’S DEFENSE IS THE THESIS THAT MATHEMATICS IS AN AESTHETIC DISCIPLINE FOR HARDY, THE APPLIED MATHEMATICS OF ENGINEERS AND ECONOMISTS HELD LITTLE CHARM “REAL MATHEMATICS,” AS HE REFERRED TO IT, “MUST BE JUSTI?ED AS ART IF IT CAN BE JUSTI?ED AT ALL” TO HELP MAKE HIS POINT, HARDY INCLUDES TWO THEOREMS FROM CLASSICAL GREEK MATHEMATICS, WHICH, IN HIS OPINION, POSSESS AN ELUSIVE KIND OF BEAUTY THAT, ALTHOUGH DI?CULT TO DE?NE, IS EASY TO RECOGNIZE THE ?RST OF THESE RESULTS IS EUCLID’S PROOF THAT THERE ARE AN IN?NITE NUMBER OF PRIME NUMBERS THE SECOND RESULT IS THE DISCOVERY, ATTRIBUTED TO THE SCHOOL OF PYTHAGORAS FROM AROUND BC, THAT ? 2 IS IRRATIONAL IT IS THIS SECOND THEOREM THAT DEMANDS OUR ATTENTION (A COURSE IN NUMBER THEORY WOULD FOCUS ON THE ?RST) THE ARGUMENT USES ONLY ARITHMETIC, BUT ITS DEPTH AND IMPORTANCE CANNOT BE OVERSTATED AS HARDY SAYS, “[IT] IS A ‘SIMPLE’ THEOREM, SIMPLE BOTH IN IDEA AND EXECUTION, BUT THERE IS NO DOUBT AT ALL ABOUT [IT BEING] OF THE HIGHEST CLASS [IT] IS AS FRESH AND SIGNI?CANT AS WHEN IT WAS DISCOVERED—TWO THOUSAND YEARS HAVE NOT WRITTEN A WRINKLE ON [IT]” THEOREM 111 THERE IS NO RATIONAL NUMBER WHOSE SQUARE IS 2 PROOF A RATIONAL NUMBER IS ANY NUMBER THAT CAN BE EXPRESSED IN THE FORM P/Q, WHERE P AND Q ARE INTEGERS THUS, WHAT THE THEOREM ASSERTS IS THAT NO MATTER HOW P AND Q ARE CHOSEN, IT IS NEVER THE CASE THAT (P/Q)2 = 2 THE LINE OF ATTACK IS INDIRECT, USING A TYPE OF ARGUMENT REFERRED TO AS A PROOF BY CONTRADICTION THE IDEA IS TO ASSUME THAT THERE IS A RATIONAL NUMBER WHOSE SQUARE IS 2 AND THEN PROCEED ALONG LOGICAL LINES UNTIL WE REACH A CONCLUSION THAT IS UNACCEPTABLE AT THIS POINT, WE WILL BE FORCED TO RETRACE OUR STEPS AND REJECT THE ERRONEOUS