Covers All 11 Chapters
,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)2and 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 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 : 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 → I 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
10.10The mapping τ2 : Q2 → I2 . . . . . 10.11The . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
mapping τ2 : Q2 → I2 . . . . . 10.12The mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Φ : D → R2 \ {0}. .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296T
10.13 ...................................................................................................................................
he spherical coordinates for the point Σ(u, v) ................................................................................................... 300
10.14 ................................................................................................................................... T
he rectangles D and E .......................................................................................................................................... 302
10.15 ................................................................................................................................... A
n example of the 2-surface S and its boundary ∂S ................................................................................ 304
10.16 ................................................................................................................................... T
he unit disk U as the projection of the unit ball V ................................................................................................ 325
10.17 ................................................................................................................................... T
he open cells U and V .......................................................................................................................................... 326
10.18 ................................................................................................................................... T
he parameter domain D......................................................................................................................................... 332
10.19 ................................................................................................................................... T
he figure of the Möbius band ................................................................................................................................ 333
10.20 .................................................................................................................................. T
he “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
,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
,List of Tables x
,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 Hölder’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 L 2 .....................................................................................................................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
Problem 1.1
Rudin Chapter 1 Exercise 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
rx
x=
x
is also rational, a contradiction. This ends the proof of the problem.
Problem 1.2
Rudin Chapter 1 Exercise 2.
√ √
Proof. Assume that 12 was rational so that 12 = m , where m and n
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
4n 2
k2 = ,
3
so n is divisible by 3. This contradicts the fact that m and n are co-prime, completing the proof of the problem.
Problem 1.3
Rudin Chapter 1 Exercise 3.
Proof. Since x 6= 0, there exists 1
∈ F such that x · 1 = 1.
x x
(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 ∈ F , we have 1
∈ F such that 1 · 1 = 1. Now we have 1 · 1 1
= 1 · x(= 1), then Proposition
x 1 x 1 x x
x x x
1
1.15(a) implies that 1 = x.
x
This completes the proof of the problem.
1.2 Properties of supremums and infimums
Problem 1.4
Rudin Chapter 1 Exercise 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.
Problem 1.5
Rudin Chapter 1 Exercise 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
Problem 1.6
Rudin Chapter 1 Exercise 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 = z qn
1 1
which implies that y = z, i.e., (bm) n = (bp ) q .
m p
(b) Let br = b n and bs = b q . Without loss of generality, we may assume that n and q are positive.
Then the corollary of Theorem 1.21 implies that
mq+np 1 p
1 11 m
mq+np mq np mq np r s
br+s =b nq = (b ) nq = (b ×b ) nq = (b ) nq × (b ) nq = b ×b = b ×b .
n q
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,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)2and 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 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 : 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 → I 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
10.10The mapping τ2 : Q2 → I2 . . . . . 10.11The . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
mapping τ2 : Q2 → I2 . . . . . 10.12The mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Φ : D → R2 \ {0}. .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296T
10.13 ...................................................................................................................................
he spherical coordinates for the point Σ(u, v) ................................................................................................... 300
10.14 ................................................................................................................................... T
he rectangles D and E .......................................................................................................................................... 302
10.15 ................................................................................................................................... A
n example of the 2-surface S and its boundary ∂S ................................................................................ 304
10.16 ................................................................................................................................... T
he unit disk U as the projection of the unit ball V ................................................................................................ 325
10.17 ................................................................................................................................... T
he open cells U and V .......................................................................................................................................... 326
10.18 ................................................................................................................................... T
he parameter domain D......................................................................................................................................... 332
10.19 ................................................................................................................................... T
he figure of the Möbius band ................................................................................................................................ 333
10.20 .................................................................................................................................. T
he “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
,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
,List of Tables x
,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 Hölder’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 L 2 .....................................................................................................................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
Problem 1.1
Rudin Chapter 1 Exercise 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
rx
x=
x
is also rational, a contradiction. This ends the proof of the problem.
Problem 1.2
Rudin Chapter 1 Exercise 2.
√ √
Proof. Assume that 12 was rational so that 12 = m , where m and n
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
4n 2
k2 = ,
3
so n is divisible by 3. This contradicts the fact that m and n are co-prime, completing the proof of the problem.
Problem 1.3
Rudin Chapter 1 Exercise 3.
Proof. Since x 6= 0, there exists 1
∈ F such that x · 1 = 1.
x x
(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 ∈ F , we have 1
∈ F such that 1 · 1 = 1. Now we have 1 · 1 1
= 1 · x(= 1), then Proposition
x 1 x 1 x x
x x x
1
1.15(a) implies that 1 = x.
x
This completes the proof of the problem.
1.2 Properties of supremums and infimums
Problem 1.4
Rudin Chapter 1 Exercise 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.
Problem 1.5
Rudin Chapter 1 Exercise 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
Problem 1.6
Rudin Chapter 1 Exercise 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 = z qn
1 1
which implies that y = z, i.e., (bm) n = (bp ) q .
m p
(b) Let br = b n and bs = b q . Without loss of generality, we may assume that n and q are positive.
Then the corollary of Theorem 1.21 implies that
mq+np 1 p
1 11 m
mq+np mq np mq np r s
br+s =b nq = (b ) nq = (b ×b ) nq = (b ) nq × (b ) nq = b ×b = b ×b .
n q
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