xii Contents 13 Consequences of the Completeness of R................. ...133 Tips:You Solved It.Now What?........................... ..140 14 Functions,Domain,and Range..................................143 Spotlight:The Definition of Function............................... 151 15 Functions,One-to-One,and Onto................................ 157 16 nverses.....167 17 Images and Inverse Images......................................181 Spotlight:Minimum or Infimum?.................................. 187 18 Mathematical Induction........................................193 19 Sequences.… 209 20 Convergence of Sequences of Real Numbers....................... 223 21 Equivalent Sets................................................ 235 22 Finite Sets and an Infinite Set....................................243 23 Countable and Uncountable Sets.................................251 24 The Cantor-Schroder-Bernstein Theorem........................ 261 Spotlight:The Continuum Hypothesis.............................. 270 25 Metric Spaces...............……… 277 26 Getting to Know Open and Closed Sets........................... 289 27 Modular Arithmetic............................................301 28Fermat's Little Theorem........................................315 Spotlight:Public and Secret Research..............................320 29 Projects.......................................................325 Tips on Talking about Mathematics................................325 29.1 Picture Proofs.............................................327 29.2 The Best Number of All (and Some Other Pretty Good Ones)......330 29.3 Set Constructions..................332 29.4 Rational and Irrational Numbers........... .334 29.5 rrationality of e andπ......… 336 29.6 A Complex Project................338 29.7 When Does f-l=1/f?..............342 29.8 Pascal's Triangle........................................... 343 29.9 The Cantor Set.......................346xii Contents 13 Consequences of the Completeness of R. . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Tips: You Solved It. Now What? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 14 Functions, Domain, and Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Spotlight: The Definition of Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 15 Functions, One-to-One, and Onto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 16 Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 17 Images and Inverse Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Spotlight: Minimum or Infimum? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 18 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 19 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 20 Convergence of Sequences of Real Numbers . . . . . . . . . . . . . . . . . . . . . . . 223 21 Equivalent Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 22 Finite Sets and an Infinite Set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 23 Countable and Uncountable Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 24 The Cantor–Schroder–Bernstein Theorem ¨ . . . . . . . . . . . . . . . . . . . . . . . . 261 Spotlight: The Continuum Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 25 Metric Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 26 Getting to Know Open and Closed Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 27 Modular Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 28 Fermat’s Little Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 Spotlight: Public and Secret Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 29 Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Tips on Talking about Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 29.1 Picture Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 29.2 The Best Number of All (and Some Other Pretty Good Ones). . . . . . 330 29.3 Set Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 29.4 Rational and Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 29.5 Irrationality of e and π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 29.6 A Complex Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 29.7 When Does f −1 = 1/ f ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 29.8 Pascal’s Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 29.9 The Cantor Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346