| Preface |
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ix | |
| The briefest overview, motivation, notation |
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xi | |
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Chapter 1 How we will do mathematics |
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1 | (44) |
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Section 1.1 Statements and proof methods |
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1 | (15) |
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Section 1.2 Statements with quantifiers |
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16 | (4) |
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Section 1.3 More proof methods |
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20 | (4) |
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Section 1.4 Logical negation |
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24 | (3) |
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27 | (3) |
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Section 1.6 Proofs by (mathematical) induction |
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30 | (10) |
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Section 1.7 Pascal's triangle |
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40 | (5) |
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Chapter 2 Concepts with which we will do mathematics |
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45 | (62) |
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45 | (11) |
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Section 2.2 Cartesian product |
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56 | (2) |
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Section 2.3 Relations, equivalence relations |
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58 | (7) |
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65 | (13) |
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Section 2.5 Binary operations |
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78 | (7) |
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85 | (5) |
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Section 2.7 Order on sets, ordered fields |
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90 | (7) |
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Section 2.8 What are the integers and the rational numbers? |
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97 | (3) |
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Section 2.9 Increasing and decreasing functions |
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100 | (3) |
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Section 2.10 Absolute values |
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103 | (4) |
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Chapter 3 Construction of the number systems |
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107 | (70) |
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Section 3.1 Inductive sets, a construction of natural numbers |
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107 | (6) |
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Section 3.2 Arithmetic on No |
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113 | (4) |
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117 | (3) |
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Section 3.4 Cancellation in No |
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120 | (1) |
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Section 3.5 Construction of Z, arithmetic, and order on Z |
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121 | (7) |
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Section 3.6 Construction of the ordered field Q |
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128 | (6) |
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Section 3.7 Construction of the field R of real numbers |
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134 | (10) |
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Section 3.8 Order on R, the Least upper bound theorem |
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144 | (6) |
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Section 3.9 Complex numbers |
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150 | (4) |
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Section 3.10 Functions related to complex numbers |
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154 | (2) |
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Section 3.11 Absolute value in C |
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156 | (4) |
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Section 3.12 Polar coordinates |
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160 | (6) |
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Section 3.13 Topology on the fields of real and complex numbers |
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166 | (5) |
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Section 3.14 The Heine-Borel theorem |
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171 | (6) |
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Chapter 4 Limits of functions |
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177 | (38) |
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Section 4.1 Limit of a function |
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177 | (13) |
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Section 4.2 When a number is not a limit |
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190 | (4) |
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Section 4.3 More on the definition of a limit |
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194 | (4) |
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Section 4.4 Limit theorems |
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198 | (9) |
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Section 4.5 Infinite limits (for real-valued functions) |
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207 | (4) |
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Section 4.6 Limits at infinity |
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211 | (4) |
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215 | (24) |
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Section 5.1 Continuous functions |
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215 | (6) |
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Section 5.2 Topology and the Extreme value theorem |
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221 | (4) |
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Section 5.3 Intermediate value theorem |
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225 | (5) |
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Section 5.4 Radical functions |
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230 | (5) |
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Section 5.5 Uniform continuity |
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235 | (4) |
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Chapter 6 Differentiation |
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239 | (30) |
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Section 6.1 Definition of derivatives |
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239 | (5) |
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Section 6.2 Basic properties of derivatives |
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244 | (9) |
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Section 6.3 The Mean value theorem |
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253 | (6) |
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Section 6.4 L'Hopital's rule |
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259 | (4) |
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Section 6.5 Higher-order derivatives, Taylor polynomials |
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263 | (6) |
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269 | (44) |
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Section 7.1 Approximating areas |
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269 | (12) |
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Section 7.2 Computing integrals from upper and lower sums |
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281 | (4) |
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Section 7.3 What functions are integrable? |
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285 | (5) |
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Section 7.4 The Fundamental theorem of calculus |
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290 | (8) |
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Section 7.5 Integration of complex-valued functions |
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298 | (1) |
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Section 7.6 Natural logarithm and the exponential functions |
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299 | (6) |
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Section 7.7 Applications of integration |
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305 | (8) |
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313 | (48) |
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Section 8.1 Introduction to sequences |
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313 | (6) |
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Section 8.2 Convergence of infinite sequences |
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319 | (9) |
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Section 8.3 Divergence of infinite sequences and infinite limits |
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328 | (4) |
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Section 8.4 Convergence theorems via epsilon-N proofs |
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332 | (7) |
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Section 8.5 Convergence theorems via functions |
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339 | (4) |
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Section 8.6 Bounded sequences, monotone sequences, ratio test |
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343 | (4) |
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Section 8.7 Cauchy sequences, completeness of R, C |
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347 | (4) |
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351 | (4) |
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Section 8.9 Liminf, limsup for real-valued sequences |
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355 | (6) |
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Chapter 9 Infinite series and power series |
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361 | (36) |
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Section 9.1 Infinite series |
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361 | (5) |
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Section 9.2 Convergence and divergence theorems for series |
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366 | (7) |
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373 | (6) |
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Section 9.4 Differentiation of power series |
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379 | (5) |
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Section 9.5 Numerical evaluations of some series |
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384 | (2) |
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Section 9.6 Some technical aspects of power series |
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386 | (5) |
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Section 9.7 Taylor series |
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391 | (6) |
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Chapter 10 Exponential and trigonometric functions |
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397 | (26) |
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Section 10.1 The exponential function |
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397 | (3) |
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Section 10.2 The exponential function, continued |
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400 | (8) |
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Section 10.3 Trigonometry |
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408 | (6) |
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Section 10.4 Examples of L'Hopital's rule |
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414 | (1) |
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Section 10.5 Trigonometry for the computation of some series |
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415 | (8) |
| Appendix A Advice on writing mathematics |
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423 | (6) |
| Appendix B What one should never forget |
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429 | (4) |
| Index |
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433 | |
