Real Analysis A Long-Form Mathematics Textbook J C ay ummings 1 -- 1/2 -- . 1/3- — 1/4-- -1 1 2 LongFormMath.com / / Contents i The Reals 1 1.1 Zeno’s Paradoxes 1 1.2 Basic Set Theory Definitions 4 1.3 What is a Number? 8 1.4 Ordered Fields 12 1.5 The Completeness Axiom . 19 1.6 Working with Sups and Infs 24 1.7 The Archimedean Principle 28 Exercises 35 2 Cardinality 43 2.1 Bijections and Cardinality . . . 43 2.2 Counting Infinities 45 2.3 How Many Infinities Arc There? 56 Exercises 60 3 Sequences 65 3.1 Basic Sequence Definitions 65 3.2 Bounded Sequences 66 3.3 Convergent Sequences 69 3.4 Divergent Sequences 3.5 Limit Laws 85 3.6 The Monotone Convergence Theorem 90 3.7 Subsequences 94 3.8 The Bolzano-Weierstrass Theorem . 99 3.9 The Cauchy Criterion ....................... 102 Exercises 110 4 Series 117 4.1 Sequences of Partial Sums 117 4.2 Series Convergence Tests . 121 4.3 Absolute Convergence . . 130 4.4 Rearrangements 134 Exercises 140 iii iv CONTENTS 5 The Topology of R 149 5.1 Open Sets 149 5.2 Closed Sets 153 5.3 Open Covers 157 5.4 The Greatest Definition in Mathematics 158 Exercises 164 6 Continuity 171 6.1 Approaching Continuity 171 6.2 Weird Examples 172 6.3 Functional Limits 177 6.4 Properties of Functional Limits . 183 6.5 Continuity 186 6.6 Topological Continuity 190 6.7 The Extreme Value Theorem . . 195 6.8 The Intermediate Value Theorem 198 6.9 Uniform Continuity 202 Exercises .................................... 208 7 Differentiation 217 7.1 Graphical Interpretations of Velocity 218 7.2 The Derivative 222 7.3 Continuity and Differentiability . . . 224 7.4 Differentiability Rules 227 7.5 Topologist’s Sine Curve Examples . . 232 7.6 Local Minimums and Maximums . . 235 7.7 The Mean Value Theorems 238 7.8 L'Hopital’s Rule .............................. 244 250 Exercises 257 8 Integration . 258 8.1 The Area of a Circle.............................. . 264 8.2 Simplistic Approach.............................. . 269 8.3 The Darboux Integral . 274 8.4 Integrability............................................. . 279 8.5 Integrability Criteria 8.6 Integrability of Continuous Functions . . . 281 8.7 Integrability of Discontinuous Functions . 284 8.8 The Measure Zero Integrability Criterion . 293 8.9 Linearity Properties of the Integral . . • . 294 8.10 More Properties of the Integral 297 8.11 The Fundamental Theorem of Calculus . 301 8.12 Integration Rules 306 Exercises 311 9 Sequences and Series of Functions 323 9.1 Introduction to Pointwise Convergence .... 323 9.2 Continuity and Functional Convergence . . . 326 331 9.3 Other Properties with Functional Convergence 334 9.4 Convergence of Derivatives and Integrals . . . 342 9.5 Series of Functions.............................................. 346 9.6 Power Series............................................................ 349 9.7 Properties of Power Series................................. 9.8 New Power Series from Old.............................. 351 9.9 Taylor and Maclaurin scries.............................. 353 9.10 A beautiful application........................................ 359 Exercises ...........................................3..6..2................ Appendices 369 A Construction of R 371 A.l Axioms of Set Theory 372 A.2 Constructing N . . . 373 A.3 Constructing Z . . . 375 A.4 Constructing Q . . . 376 A.5 Constructing R . . . 377 B Peculiar and Pathological Examples 379 B.l The Curious Case of the Cantor Set . . . 381 B.2 Doubled Digits of Diametrical Degrees . . 384 B.3 Structuring Stuff from its Shadows’ Shapes 386 B.4 Monger’s Matterless Material 388 B.5 Obtainable Outrageousness in an Orderly Overhang 391 B.6 A Composition Conundrum . 396 B.7 Turning the Tables on your Teetering Troubles . . 398 B.8 A Devilish De-Descent.......................................... 400 B.9 A Pack of Pretty Proofs by Picture.......................... 404 B.10 An Abundant Addition Aboundingly Ascends . . . 407 B.ll A Smooth and Spiky Solution.................................... 408 B.l2 Finding </> for First-Place Finishes.............................. 412 B.l3 Peculiar and Pathological Perimeters....................... 414 B.14 Fractal Functions Filling Foursquare Frames . . . . 416 B.15 A Prestigious Proof of a Primal Puzzle.................... 419 B.l6 A Topological Treatment with a Tremendous Twist 422 B.l7 Modern Measuring’s Misfit Member.......................... 424 B.18 Tarski’s Terrific Talents Times Two.......................... 426 Index 429 List of Results 15 1.12 Proposition (|<z| < b iff — b < a < b) 15 1.13 Theorem (The triangle inequality) 17 1.14 Corollary (The reverse triangle inequality) . ■ ■ 18 1.15 Corollary (Triangle inequality corollaries) . . . 22 1.20 Theorem (Existence and uniqueness ofW) ... 24 1.22 Proposition (Supreme are unique) 25 1.23 Theorem (Square roots exist) 26 1.24 Theorem (Supreme analytically) 28 1.26 Lemma (The Archimedean principle) 1.30 Lemma (y—x>l=>3z&Z where x < z < y) 30 1.31 Theorem (Q is dense in R) 31 1.34 Principle (Well-ordering principle) 33 2.1 Principle (The bijection principle) 43 2.8 Theorem (|Z| = |Q|) 47 2.9 Theorem (|R| > |N|) 50 2.11 Theorem (Sizes of infinity) 53 2.13 Theorem (|/| < |7?(/1)|) 56 2.14 Corollary (There exist infinitely many infinities) 57 3.5 Proposition (Bounded O |a„| < C) 67 3.19 Proposition (Limits are unique) 81 3.20 Proposition (Convergent => bounded) 83 3.21 Theorem (Sequence limit laws) 85 3.23 Theorem (Sequence squeeze theorem) 87 3.27 Theorem (The monotone convergence theorem) . . . 90 3.29 Proposition (Bounded S contains at, —> sup(S)) . . . 93 3.32 Proposition (an —> a <=> every allk -> a) 95 3.34 Corollary (Different subsequential limits => diverges) 96 3.35 Proposition (Monotone (an) has ank —> a => an —> a) 97 3.36 Lemma ((an) has monotone allk) 99 3.37 Theorem (The Bolzano-Weierstrass theorem) . . . . 101 3.41 Lemma (Cauchy => bounded) 106 3.42 Theorem (Cauchy criterion for convergence) 107 vii viii LIST OF RESULTS 4.3 Corollary (Series (limit) laws) 119 4.5 Proposition (k-term test) 121 4.8 Lemma ((r") converges iff r G (-1,1]) 122 4.9 Proposition (Geometric series test) . . 123 4.11 Proposition (a* > 0 => converges or = oo) 124 4.12 Proposition (Comparison test) 125 4.15 Proposition I diverges) 126 4.16 Proposition (The series p-test) .... 128 4.17 Proposition (Alternating series test) . 130 4.18 Proposition (^ |a*| converges => converges) 132 4.23 Theorem (Rearrangement theorem) 134 5.3 Proposition (Open sets via arbitrary unions and finite intersections) 151 5.5 Theorem (Each open set = U£i(afc>M) 153 5.10 Theorem (Closed iff contains limit points) 154 5.12 Proposition (Closed sets via finite unions and arbitrary intersections) 156 5.19 Theorem (The Heine-Borel theorem) 159 5.20 Theorem (Heine-Borel, expanded) 162 6.11 Proposition (Functional limits are unique)............ 183 6.12 Theorem (lim f(x) = L iff every such f(an) -> L) . .......................... 184 x-+c 6.14 Corollary (Func-y limit laws)................................. 185 6.15 Corollary (Func-y squeeze theorem)........................ 186 6.17 Theorem (Continuity topologically and sequentially) 187 6.20 Proposition (Continuity limit laws)........................ 188 6.24 Proposition (fig continuous => f o g is too.) .... .......................... 189 6.29 Theorem (f continuous <=> B open implies f~1(B) - open Q X) . . 192 6.30 Theorem (The continuous image of a compact set is compact) . . . 195 6.31 Corollary (Continuous on compact set => bounded) 196 6.32 Theorem (The extreme value theorem) 197 6.36 Lemma (f continuous, f(c) > 0 => f(x) > 0 V |a: - c| < 6) 198 6.37 Proposition (f continuous, f(a) > 0 > f(b) => 3 c 6 (a, b), f(c) =0) 199 6.38 Theorem (The intermediate value theorem) ......................................... 200 6.40 Proposition (Continuous f on compact A => f uniformly continuous') 205 7.6 Theorem (Differentiable => continuous) . 225 7.9 Proposition (Linearity of the derivative) • 227 7.11 Theorem (The product rule) ■ 228 7.12 Theorem (The quotient rule) ■ 229 7.13 Theorem (The chain rule) • 230 7.19 Proposition (Local maxes/mins happen where f(c) =0) ... • 235 7.20 Theorem (Darboux’s theorem) • 237 7.21 Theorem (Rolle’s theorem) • 239 7.22 Theorem (The (derivative) mean value theorem) • 241 7.23 Corollary (f(x) — 0 => f(x) = C) • 242 / 7.24 Corollary (“Don't forget your +CI”) . . 242 7.25 Corollary (/' > 0 O f is increasing) . 243 7.26 Theorem (Cauchy mean value theorem) 245 7.27 Theorem (L’Hopital’s rule) 245 8.6 Proposition (Refinements refine) 271 8.7 Proposition (Lower sums < Upper sums) 273 8.9 Lemma (m(b — a) < L(f) < U(f) < M(b — a)) 274 8.13 Proposition (f > 0 => J f > 0)................................................. 278 8.14 Theorem (Integrals analytically)................................................. 280 8.15 Corollary (f integrable =4- lim[J7(/, Pn) — L(/, Pn)] = 0) . . 281 8.16 Theorem (Continuous => integrable) 283 8.18 Porism (fa integrable for any discontinuity c) 286 8.19 Lemma (f integrable on [a, c] and [c, 6] <=> on [a, &]) 287 8.24 Theorem (f integrable <=> discontinuities have measure zero) 294 8.25 Proposition (fr, f = J{, f + J? f) 294 8.26 Proposition (Linearity of the integral) 296 8.28 Corollary (f < g => / f < f g) 297 8.29 Corollary (| f f\ < f \f\) . 298 8.31 Proposition (The integral mean value theorem) 300 8.32 Theorem (The fundamental theorem of calculus) 302 8.34 Corollary (Integration by parts)................................................. 306 8.35 Corollary (w-substitution) 307 9.8 Proposition (fa continuous, fa —> f uniformly => f continuous) . . . 330 9.10 Proposition (fa bounded, fa -> f uniformly => f bounded) 332 9.12 Proposition (fa unif. cts., fa —> f uniformly => f unif. cts.)............. 333 9.16 Proposition (fa integrable, fa-> f uniformly => f integrable) . . . . 337 9.17 Proposition (Uniform limit of differentiable functions) . . . 339 9.18 Proposition (Cauchy criterion for sequences of functions) . 340 9.19 Theorem (The Arzela-Ascoli theorem) 341 9.21 Proposition (Cauchy criterion for series of functions) . . . . . . 343 9.22 Corollary ( Weierstrass M-test).................................................. . . 344 9.26 Lemma (P.S. converges on (—xo,xq) => unif. on closed subintervals) 347 9.27 Theorem (P.S. converge on an interval centered at 0) . . . . . . 348 9.30 Theorem (P.S. converges absolutely at c => uniformly on [-c,c]) . . 349 9.31 Theorem (Derivatives and integrals of P.S.) 350 9.38 Lemma (f equals its Taylor series <=> JSn -> 0)................ 356 9.39 Theorem (Integral form of the error function)................ 356 9.40 Theorem (Lagrange form of the error function) 358 I