I am also very grateful to the anonymous referees who made several correc- tions and suggested many important improvements to the text Terence Tao
Texts and Readings in Mathematics 37 Analysis I Third Edition Terence Tao ISBN 978-981-10-1789-6 (eBook) the relevant protective laws and regulations and therefore free for general use B 2 The decimal representation of real numbers as 0 = 1 This motivated the need to go back to the very beginning of the
I am also very grateful to the anonymous referees who made several correc- tions and suggested many important improvements to the text Terence Tao
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Terence Tao Department of Mathematics University of California, Los Angeles Los Angeles, CA USA This work is a co-publication with Hindustan Book
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Contents Preface x 1 Introduction 1 1.1 What is analysis? . . . . . . . . . . . . . . . . . . . 1 1.2 Why do analysis? . . . . . . . . . . . . . . . . . . . 3 2 The natural numbers 14 2.1 The Peano axioms . . . . . . . . . . . . . . . . . . 16 2.2 Addition . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3 Multiplication . . . . . . . . . . . . . . . . . . . . . 33 3 Set theory 37 3.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . 37 3.2 Russell's paradox (Optional) . . . . . . . . . . . . 52 3.3 Functions . . . . . . . . . . . . . . . . . . . . . . . 55 3.4 Images and inverse images . . . . . . . . . . . . . . 64 3.5 Cartesian products . . . . . . . . . . . . . . . . . . 70 3.6 Cardinality of sets . . . . . . . . . . . . . . . . . . 78 4 Integers and rationals 85 4.1 The integers . . . . . . . . . . . . . . . . . . . . . . 85 4.2 The rationals . . . . . . . . . . . . . . . . . . . . . 93 4.3 Absolute value and exponentiation . . . . . . . . . 99 4.4 Gaps in the rational numbers . . . . . . . . . . . . 104 5 The real numbers 108 5.1 Cauchy sequences . . . . . . . . . . . . . . . . . . . 110 viCONTENTS
5.2 Equivalent Cauchy sequences . . . . . . . . . . . . 115 5.3 The construction of the real numbers . . . . . . . . 118 5.4 Ordering the reals . . . . . . . . . . . . . . . . . . 128 5.5 The least upper bound property . . . . . . . . . . 134 5.6 Real exponentiation, part I . . . . . . . . . . . . . 140 6 Limits of sequences 146 6.1 The Extended real number system . . . . . . . . . 154 6.2 Suprema and In¯ma of sequences . . . . . . . . . . 158 6.3 Limsup, Liminf, and limit points . . . . . . . . . . 161 6.4 Some standard limits . . . . . . . . . . . . . . . . . 171 6.5 Subsequences . . . . . . . . . . . . . . . . . . . . . 172 6.6 Real exponentiation, part II . . . . . . . . . . . . . 176 7 Series 179 7.1 Finite series . . . . . . . . . . . . . . . . . . . . . . 179 7.2 In¯nite series . . . . . . . . . . . . . . . . . . . . . 189 7.3 Sums of non-negative numbers . . . . . . . . . . . 195 7.4 Rearrangement of series . . . . . . . . . . . . . . . 200 7.5 The root and ratio tests . . . . . . . . . . . . . . . 204 8 In¯nite sets 208 8.1 Countability . . . . . . . . . . . . . . . . . . . . . . 208 8.2 Summation on in¯nite sets . . . . . . . . . . . . . . 216 8.3 Uncountable sets . . . . . . . . . . . . . . . . . . . 224 8.4 The axiom of choice . . . . . . . . . . . . . . . . . 228 8.5 Ordered sets . . . . . . . . . . . . . . . . . . . . . . 232 9 Continuous functions on R 243 9.1 Subsets of the real line . . . . . . . . . . . . . . . . 244 9.2 The algebra of real-valued functions . . . . . . . . 251 9.3 Limiting values of functions . . . . . . . . . . . . . 254 9.4 Continuous functions . . . . . . . . . . . . . . . . . 262 9.5 Left and right limits . . . . . . . . . . . . . . . . . 267 9.6 The maximum principle . . . . . . . . . . . . . . . 270 9.7 The intermediate value theorem . . . . . . . . . . . 275 9.8 Monotonic functions . . . . . . . . . . . . . . . . . 277 CONTENTSvii 9.9 Uniform continuity . . . . . . . . . . . . . . . . . . 280 9.10 Limits at in¯nity . . . . . . . . . . . . . . . . . . . 287 10 Di®erentiation of functions 290 10.1 Local maxima, local minima, and derivatives . . . 297 10.2 Monotone functions and derivatives . . . . . . . . . 300 10.3 Inverse functions and derivatives . . . . . . . . . . 302 10.4 L'H^opital's rule . . . . . . . . . . . . . . . . . . . . 305 11 The Riemann integral 308 11.1 Partitions . . . . . . . . . . . . . . . . . . . . . . . 309 11.2 Piecewise constant functions . . . . . . . . . . . . . 314 11.3 Upper and lower Riemann integrals . . . . . . . . . 318 11.4 Basic properties of the Riemann integral . . . . . . 323 quotesdbs_dbs7.pdfusesText_5
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