, 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, photo-
copying, 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), N h (x) and Nqm (xk ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2
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, π2 ]. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 X 2 + 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 : I 2 → 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 : I 3 → Q3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
10.9 The mapping τ1 : Q2 → I 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
10.10The mapping τ2 : Q2 → I 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
10.11The mapping τ2 : Q2 → I 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
10.12The mapping Φ : D → R2 \ {0}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
10.13The spherical coordinates for the point Σ(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 Möbius 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 Ω measured in steradians. . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
B.3 A section of the cone with apex angle 2θ. . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
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, photo-
copying, 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), N h (x) and Nqm (xk ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2
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, π2 ]. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 X 2 + 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 : I 2 → 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 : I 3 → Q3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
10.9 The mapping τ1 : Q2 → I 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
10.10The mapping τ2 : Q2 → I 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
10.11The mapping τ2 : Q2 → I 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
10.12The mapping Φ : D → R2 \ {0}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
10.13The spherical coordinates for the point Σ(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 Möbius 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 Ω measured in steradians. . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
B.3 A section of the cone with apex angle 2θ. . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
