Exam (elaborations) TEST BANK FOR Principles of Mathematical Analysis By Walter Rudin (A Complete Solution Guide)

Exam (elaborations) TEST BANK FOR Principles of Mathematical Analysis By Walter Rudin (A Complete Solution Guide) A Complete Solution Guide to Principles of Mathematical Analysis by Kit-Wing Yu, PhD Copyright c 2018 by Kit-Wing Yu. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the author. ISBN: 978-988-78797-0-1 (eBook) ISBN: 978-988-78797-1-8 (Paperback) List of Figures 2.1 The neighborhoods Nh(q) and Nr(p). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Convex sets and nonconvex sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 The sets Nh(x),Nh 2 (x) and Nqm(xk). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4 The construction of the shrinking sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.1 The Cantor set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.1 The graph of g on [an, bn]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2 The sets E and Ini . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.3 The graphs of [x] and (x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.4 An example for α = √2 and n = 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.5 The distance from x ∈ X to E. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.6 The graph of a convex function f. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.7 The positions of the points p, p + κ, q − κ and q. . . . . . . . . . . . . . . . . . . . . . . . 77 5.1 The zig-zag path of the process in (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.2 The zig-zag path induced by the function f in Case (i). . . . . . . . . . . . . . . . . . . . 108 5.3 The zig-zag path induced by the function g in Case (i). . . . . . . . . . . . . . . . . . . . 109 5.4 The zig-zag path induced by the function f in Case (ii). . . . . . . . . . . . . . . . . . . 109 5.5 The zig-zag path induced by the function g in Case (ii). . . . . . . . . . . . . . . . . . . 110 5.6 The geometrical interpretation of Newton’s method. . . . . . . . . . . . . . . . . . . . . . 111 8.1 The graph of the continuous function y = f(x) = (π − |x|)2 on [−π, π]. . . . . . . . . . . . 186 8.2 The graphs of the two functions f and g. . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 8.3 A geometric proof of 0 < sin x ≤ x on (0, π 2 ]. . . . . . . . . . . . . . . . . . . . . . . . . . . 199 8.4 The graph of y = | sin x|. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 8.5 The winding number of γ around an arbitrary point p. . . . . . . . . . . . . . . . . . . . . 202 8.6 The geometry of the points z, f(z) and g(z). . . . . . . . . . . . . . . . . . . . . . . . . . . 209 9.1 An example of the range K of f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 9.2 The set of q ∈ K such that (∇f3)(f−1(q)) = 0. . . . . . . . . . . . . . . . . . . . . . . . . 220 9.3 Geometric meaning of the implicit function theorem. . . . . . . . . . . . . . . . . . . . . . 232 9.4 The graphs around the four points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 9.5 The graphs around (0, 0) and (1, 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 9.6 The graph of the ellipse X2 + 4Y 2 = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 9.7 The definition of the function ϕ(x, t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 9.8 The four regions divided by the two lines αx1 + βx2 = 0 and αx1 − βx2 = 0. . . . . . . . 252 10.1 The compact convex set H and its boundary ∂H. . . . . . . . . . . . . . . . . . . . . . . . 256 10.2 The figures of the sets Ui,Wi and Vi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 10.3 The mapping T : I2 → H. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 10.4 The mapping T : A → D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 10.5 The mapping T : A◦ → D0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 10.6 The mapping T : S → Q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 vii List of Figures viii 10.7 The open sets Q0.1,Q0.2 and Q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 10.8 The mapping T : I3 → Q3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 10.9 The mapping τ1 : Q2 → I2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 10.10The mapping τ2 : Q2 → I2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 10.11The mapping τ2 : Q2 → I2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 10.12The mapping  : D → R2 {0}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 10.13The spherical coordinates for the point (u, v). . . . . . . . . . . . . . . . . . . . . . . . . 300 10.14The rectangles D and E. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 10.15An example of the 2-surface S and its boundary ∂S. . . . . . . . . . . . . . . . . . . . . . 304 10.16The unit disk U as the projection of the unit ball V . . . . . . . . . . . . . . . . . . . . . . 325 10.17The open cells U and V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 10.18The parameter domain D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 10.19The figure of the M¨obius band. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 10.20The “geometric” boundary of M. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 11.1 The open square Rδ((p, q)) and the neighborhood N√2δ((p, q)). . . . . . . . . . . . . . . . 350 B.1 The plane angle θ measured in radians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 B.2 The solid angle measured in steradians. . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 B.3 A section of the cone with apex angle 2θ. . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 List of Tables 6.1 The number of intervals & end-points and the length of each interval for each En. . . . . 121 9.1 Expressions of x around four points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 9.2 Expressions of y around four points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 ix Contents Preface v List of Figures vii List of Tables ix 1 The Real and Complex Number Systems 1 1.1 Problems on rational numbers and fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Properties of supremums and infimums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 An index law and the logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.4 Properties of the complex field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Properties of Euclidean spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.6 A supplement to the proof of Theorem 1.19 . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Basic Topology 11 2.1 The empty set and properties of algebraic numbers . . . . . . . . . . . . . . . . . . . . . . 11 2.2 The uncountability of irrational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Limit points, open sets and closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Some metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.5 Compact sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.6 Further topological properties of R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.7 Properties of connected sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.8 Separable metric spaces and bases and a special case of Baire’s theorem . . . . . . . . . . 24 3 Numerical Sequences and Series 31 3.1 Problems on sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 Problems on series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Recursion formulas of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.4 A representation of the Cantor set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.5 Cauchy sequences and the completions of metric spaces . . . . . . . . . . . . . . . . . . . 50 4 Continuity 57 4.1 Properties of continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2 The extension, the graph and the restriction of a continuous function . . . . . . . . . . . . 58 4.3 Problems on uniformly continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.4 Further properties of continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.5 Discontinuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.6 The distance function ρE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.7 Convex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.8 Other properties of continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 xi Contents 5 Differentiation 85 5.1 Problems on differentiability of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2 Applications of Taylor’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.3 Derivatives of higher order and iteration methods . . . . . . . . . . . . . . . . . . . . . . . 102 5.4 Solutions of differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6 The Riemann-Stieltjes Integral 117 6.1 Problems on Riemann-Stieltjes integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.2 Definitions of improper integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.3 H¨older’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.4 Problems related to improper integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.5 Applications and a generalization of integration by parts . . . . . . . . . . . . . . . . . . . 133 6.6 Problems on rectifiable curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7 Sequences and Series of Functions 141 7.1 Problems on uniform convergence of sequences of functions . . . . . . . . . . . . . . . . . 141 7.2 Problems on equicontinuous families of functions . . . . . . . . . . . . . . . . . . . . . . . 157 7.3 Applications of the (Stone-)Weierstrass theorem . . . . . . . . . . . . . . . . . . . . . . . . 164 7.4 Isometric mappings and initial-value problems . . . . . . . . . . . . . . . . . . . . . . . . . 167 8 Some Special Functions 173 8.1 Problems related to special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 8.2 Index of a curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 8.3 Stirling’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 9 Functions of Several Variables 213 9.1 Linear transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 9.2 Differentiable mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 9.3 Local maxima and minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 9.4 The inverse function theorem and the implicit function theorem . . . . . . . . . . . . . . . 226 9.5 The rank of a linear transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 9.6 Derivatives of higher order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 10 Integration of Differential Forms 255 10.1 Integration over sets in Rk and primitive mappings . . . . . . . . . . . . . . . . . . . . . . 255 10.2 Generalizations of partitions of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 10.3 Applications of Theorem 10.9 (Change of Variables Theorem) . . . . . . . . . . . . . . . . 267 10.4 Properties of k-forms and k-simplexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 10.5 Problems on closed forms and exact forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 10.6 Problems on vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 11 The Lebesgue Theory 337 11.1 Further properties of integrable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 11.2 The Riemann integrals and the Lebesgue integrals . . . . . . . . . . . . . . . . . . . . . . 340 11.3 Functions of classes L and L2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 Appendix 355 A A proof of Lemma 10.14 355 B Solid angle subtended by a surface at the origin 365 C Proofs of some basic properties of a measure 369 Index 377 Bibliography 379 CHAPTER 1 The Real and Complex Number Systems Unless the contrary is explicitly stated, all numbers that are mentioned in these exercises are understood to be real. 1.1 Problems on rational numbers and fields Rudin Chapter 1 Exercise 1. Problem 1.1 Proof. Assume that r + x was rational. Then it follows from Definition 1.12(A1), (A4) and (A5) that x = (r + x) − x is also rational, a contradiction. Similarly, if rx was rational, then it follows from Definition 1.12(M1), M(4) and M(5) that x = rx x is also rational, a contradiction. This ends the proof of the problem.  Rudin Chapter 1 Exercise 2. Problem 1.2 Proof. Assume that √12 was rational so that √12 = m n , where m and n are co-prime integers. Then we have m2 = 12n2 and thus m is divisible by 3. Let m = 3k for some integer k. Then we have m2 = 9k2 and this shows that k2 = 4n2 3 , so n is divisible by 3. This contradicts the fact that m and n are co-prime, completing the proof of the problem.  Rudin Chapter 1 Exercise 3. Problem 1.3 Proof. Since x 6= 0, there exists 1 x ∈ F such that x · 1 x = 1. (a) Therefore, it follows from Definition 1.12(M2), (M3), (M4) and (M5) that xy = xz implies that y = z. 1 Chapter 1. The Real and Complex Number Systems 2 (b) Similarly, it follows from Definition 1.12(M2), (M3), (M4) and (M5) that xy = x implies that y = 1. (c) Similarly, it follows from Definition 1.12(M2), (M3), (M4) and (M5) that xy = 1 implies that y = 1 x. (d) Since 1 x ∈ F, we have 1 1 x ∈ F such that 1 x · 1 1 x = 1. Now we have 1 x · 1 1 x = 1 x ·x(= 1), then Proposition 1.15(a) implies that 1 1 x = x. This completes the proof of the problem.  1.2 Properties of supremums and infimums Rudin Chapter 1 Exercise 4. Problem 1.4 Proof. Since E ⊂ S, the definitions give α ≤ x and x ≤ β for all x ∈ E. Thus Definition 1.5(ii) implies that α ≤ β. This completes the proof of the problem.  Rudin Chapter 1 Exercise 5. Problem 1.5 Proof. Theorem 1.19 says that R is an ordered set with the least-upper-bound property. Since A is a non-empty subset of R and A is bounded below, inf A exists in R by Definition 1.10. Furthermore, −A is a non-empty subset of R. Let y be a lower bound of A, i.e. y ≤ x for all x ∈ A. Then we have −x ≤ −y for all x ∈ A. Thus −y is an upper bound of −A and sup(−A) exists in R by Definition 1.10. Let α = inf A and β = sup(−A). By definition, we have y ≤ β for all y ∈ −A, where y = −x for some x ∈ A. It implies that x = −y ≥ −β for all x ∈ A, so −β is a lower bound of A and then −β ≤ α. Similarly, we have α ≤ x for all x ∈ A so that −α ≥ −x for all x ∈ A. It implies that −α is an upper bound of −A, so β ≤ −α and then −β ≥ α. Hence we have α = −β, i.e. inf A = −sup(−A). This completes the proof of the problem.  1.3 An index law and the logarithm Rudin Chapter 1 Exercise 6. Problem 1.6 Proof. (a) Since bm > 0 and n ∈ N, Theorem 1.21 implies that there exists one and only one real y such that yn = bm. Similarly, there exists one and only one real z such that zq = bp. We have ynq = (yn)q = (bm)q = bmq = bpn = (bp)n = (zq)n = zqn which implies that y = z, i.e., (bm) 1 n = (bp) 1 q . (b) Let br = b m n and bs = b p q . Without loss of generality, we may assume that n and q are positive. Then the corollary of Theorem 1.21 implies that br+s = b mq+np nq = (bmq+np) 1 nq = (bmq × bnp) 1 nq = (bmq) 1 nq × (bnp) 1 nq = b m n × b p q = br × bs. 3 1.3. An index law and the logarithm (c) By definition, B(r) = {bt | t ∈ Q, t ≤ r}, where r ∈ Q. It is clear that br ∈ B(r), so it is a nonempty subset of R. Since b > 1, we have bt ≤ br for all t ≤ r so that br is an upper bound of B(r). Therefore, Theorem 1.19 and Definition 1.10 show that supB(r) exists in R. Now we show that br = supB(r). If 0 < γ < br, then γ is obviously not an upper bound of B(r) because br ∈ B(r). By Definition 1.8, we have br = supB(r). (d) By part (c), we know that bx, by and bx+y are all well-defined in R. By definition, we have B(x) = {br | r ∈ Q, r ≤ x}, B(y) = {bs | s ∈ Q, s ≤ y}, B(x + y) = {bt | t ∈ Q, t ≤ x + y}. Before continuing the proof, we need to show several results: For every real x and y, we define B(x, y) = B(x) × B(y) = {br × bs | r, s ∈ Q, r ≤ x, s ≤ y}. Then we have bx × by = supB(x, y). Lemma 1.1 Proof of Lemma 1.1. By definition, bx and by are upper bounds of B(x) and B(y) respectively, so we have br ≤ bx and bs ≤ by for every br ∈ B(x) and bs ∈ B(y). Therefore, we have br × bs ≤ bx × by for every br × bs ∈ B(x, y). In other words, bx × by is an upper bound of B(x, y). Let 0 < α < bx × by. Then we have α bx < by. We define the number p = 1 2 ( α bx + by). It is obvious from this definition that α bx < p < by. By α bx < p, we have α p < bx and so there exists br ∈ B(x) such that α p < br. (1.1) Similarly, the inequality p < by implies that there exists bs ∈ B(y) such that p < bs. (1.2) Now inequalities (1.1) and (1.2) show that α < br × bs for some br × bs ∈ B(x, y). Hence α is not an upper bound of B(x, y) and we have bx × by = supB(x, y), completing the proof of the lemma.  Let S be a set of positive real numbers and bounded above and S−1 = {x−1 | x ∈ S}. Then we have sup S = 1 inf S−1 . Lemma 1.2 Proof of Lemma 1.2. Suppose that α is an upper bound of S, i.e., 0 < x ≤ α for all x ∈ S. Then we have 0 < α−1 ≤ x−1 for all x−1 ∈ S−1. Hence the result follows directly from the definitions of the least upper bound and the greatest lower bound.  Chapter 1. The Real and Complex Number Systems 4 For every real x, we have b−x = 1 bx . Lemma 1.3 Proof of Lemma 1.3. We have two facts: – Fact 1. If b > 1, then b 1 n > 1 for every positive integer n. Otherwise, 0 < b 1 n < 1 implies that 0 < b = (b 1 n )n < 1n = 1 by Theorem 1.21, a contradiction. – Fact 2. If m and n are positive integers such that n > m, then b 1 m > b 1 n . Otherwise, it follows from Fact 1 that 1 < b 1 m < b 1 n and so it implies that b = (b 1 m )m < (b 1 n )m < (b 1 n )n = b, a contradiction. Let r and s be rational. Define A(x) = {bs | s ∈ Q, s ≥ x}. We next want to prove that supB(x) = inf A(x). In fact, it is clear that supB(x) ≤ inf A(x) by definitions. Suppose that D = inf A(x) − supB(x). Assume that D > 0. By Fact 2 above, b 1 n − 1 is decreasing as n is increasing, so there exists a positive integer N such that bx(b 1 n − 1) < D for all n ≥ N. By Theorem 1.20(b), we see that there exist r, s ∈ Q such that x − 1 2n < s < x and x < r < x + 1 2n. Thus we have r − s < x + 1 2n − (x − 1 2n) = 1 n and this implies that D < br − bs = bs(br−s − 1) ≤ bx(br−s − 1) < D, a contradiction. Hence we must have supB(x) = inf A(x). Define B′(−x) = {b−t | t ∈ Q, t ≤ −x}. Let s = −t. Then we have b−t = bs and the inequality t ≤ −x is equivalent to s ≥ x. Thus we have B′(−x) = A(x). Now it follows from Lemma 1.2 and the above facts that b−x = supB(−x) = 1 inf B′(−x) = 1 inf A(x) = 1 supB(x) = 1 bx , completing the proof of the lemma.  We can continue our proof of the problem. Let br and bs be any elements of B(x) and B(y) respectively. Then we have r and s are rational numbers such that r ≤ x and s ≤ y. Since the sum t = r + s is also rational and t ≤ x + y, part (b) implies that br × bs = br+s ≤ bx+y. Thus bx+y is an upper bound of the set B(x, y) and Lemma 1.1 implies that bx × by ≤ bx+y. Assume that bxby < bx+y. We deduce from Lemma 1.3 that by = (b−xbx)by = b−x(bxby) < b−xbx+y ≤ b−x+(x+y) = by, a contradiction. Hence we have the desired result that bxby = bx+y. This completes the proof of the problem.  Rudin Chapter 1 Exercise 7. Problem 1.7 Proof. 5 1.4. Properties of the complex field (a) Since b > 1, it is obvious that bn − 1 = (b − 1)(bn−1 + bn−2 + · · · + 1) ≥ (b − 1)(1 + 1 + · · · + 1) = n(b − 1) for any positive integer n. (b) It is obvious that b 1 n > 1; otherwise, we have b = (b 1 n )n < 1 which is impossible. The result follows by replacing b by the real number b 1 n in part (a) and Problem 1.6(a). (c) If t > 1 and n > b−1 t−1 , then part (b) implies that b − 1 ≥ n(b 1 n − 1) > b − 1 t − 1 × (b 1 n − 1) and so b 1 n < t. (d) Let w be a number such that bw < y. Let t = y · b−w. It is easily to check that t > 1. If n is sufficiently large enough, then we have n > b−1 t−1 . Hence it follows from parts (c) and (b) that b 1 n < t = y · b−w and thus bw+ 1 n < y for sufficiently large n. (e) Let w be a number such that bw > y. Let t = y−1 · bw. It is obvious that t > 1. If n is sufficiently large enough, then we have n > b−1 t−1 . Hence it follows from part (c) and then part (b) that n(y−1bw − 1) > b − 1 ≥ n(b 1 n − 1) and thus bw− 1 n > y for sufficiently large n. (f) We have A = {w ∈ R| bw < y}. Since x is the least upper bound of A, we have w ≤ x for all w ∈ A. If bx < y, then part (d) implies that bx+ 1 n < y for sufficiently large n and so x + 1 n ∈ A. Therefore, we have x + 1 n ≤ x and then 1 n ≤ 0, a contradiction. Similarly, if bx > y, then x /∈ A and so w < x for all w ∈ A. Now part (e) implies that bx− 1 n > y for sufficiently large n, so we have w < x − 1 n < x for some sufficiently large n. This means that x − 1 n is an upper bound of A, contradicting to the fact that x is the least upper bound of A. Hence, we must have bx = y. (g) The uniqueness of x follows from the uniqueness of the least upper bound of the set A. We finish the proof of the problem.  1.4 Properties of the complex field Rudin Chapter 1 Exercise 8. Problem 1.8 Proof. Assume that i > 0. Then we have i ·i > i ·0 which implies that −1 > 0, a contradiction. Similarly, the case i < 0 is impossible.  Rudin Chapter 1 Exercise 9. Problem 1.9