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5.2 The Archimedean Property 169 EXERCISE 5.1.10 The greatest lower bound of a set is unique. The next exercise will serve as a helpful lemma for the exercise to follow. EXERCISE 5.1.11 Suppose A is a set that is bounded from below by M, and let B ={−a : a ∈ A}. Then B is bounded from above by −M. EXERCISE 5.1.12 If A is a non-empty set of real numbers that is bounded from below, then A has a GLB. 2 EXERCISE 5.1.13 State and prove a theorem analogous to Theorem 5.1.5 for the GLB of a set. EXERCISE 5.1.14 Let A be a subset of the real numbers, and define the set B ={−a : a ∈ A}. Then A has LUB L if and only if B has GLB −L. 3 5.2 The Archimedean Property The LUB property implies that the positive integers are unbounded from above, which is the claim of the next exercise. To prove it, suppose the result is false, claim the existence of a LUB, and then exploit properties M1–M2 for a strategically chosen value of ɛ to produce a contradiction. EXERCISE 5.2.1 For all positive real numbers x, there exists a positive integer n such that n>x. Exercise 5.2.1 has a logically equivalent form called the Archimedean property. We have already pointed out briefly that the ancient Greeks were very sophisticated mathematically. One idea they used involved what we would call an infinitesimal number. Different from zero, an infinitesimal was considered to be smaller than every positive number. One salient property of an infinitesimal is that you can add it to itself any finite number of times and still have an infinitesimal. The next result says that there are no infinitesimals in the real numbers. It is named after Archimedes, who argued against the use of infinitesimals. EXERCISE 5.2.2 [Archimedean Property] For all positive real numbers ɛ and r, there exists a positive integer n such that nɛ>r. 4 Thus no matter how small a positive number ɛ might be, and no matter how large a positive number r is, you can add ɛ to itself some n number of times 2 If B from Exercise 5.1.11 has LUB L, you can show that −L is the GLB for A. 3 You might be able to appeal to your proof of Exercise 5.1.12. 4 Apply Exercise 5.2.1 to x = r/ɛ.
170 Chapter 5 The Real Numbers to produce a sum that exceeds r, so that ɛ is therefore not an infinitesimal. The Archimedean property often proves to be useful in a slightly different form, using the specific case r = 1. We state it as a corollary to Exercise 5.2.2. Corollary 5.2.3 For all ɛ>0, there exists a positive integer n such that 1/n
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A Transition to Abstract Mathematic
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A Transition to Abstract Mathematic
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For Topo my little mouse
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Contents Why Read This Book? xiii P
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3.10 Irrational Numbers 107 3.11 Re
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8.2 Subgroups 252 8.2.1 Subgroups D
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Why Read This Book? One of Euclid
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Preface A Transition to Abstract Ma
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Preface to the First Edition This t
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Preface to the First Edition xix ea
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Acknowledgments It takes an entire
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Notation and Assumptions 0 Suppose
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0.2 Assumptions about the Real Numb
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0.2 Assumptions about the Real Numb
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0.2.3 Other Assumptions 0.2 Assumpt
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P A R T I Foundations of Logic and
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Language and Mathematics 1 One main
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1.1 Introduction to Logic 13 Now im
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The compound statement we call “p
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1.1 Introduction to Logic 17 exampl
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1.2 If-Then Statements 19 Definitio
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1.2 If-Then Statements 21 (h) The o
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1.2 If-Then Statements 23 (g) If ei
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p : Meghan is at least 25 years old
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1.3 Universal and Existential Quant
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1.4 Negations of Statements 33 2. F
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1.4 Negations of Statements 35 want
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Solution 1.4 Negations of Statement
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Solution 1. Everyone in this class
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1.5 How We Write Proofs 41 Our purp
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1.5 How We Write Proofs 43 the nega
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Properties of Real Numbers 2 It’s
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2.1 Basic Algebraic Properties of R
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2.1.2 Properties of Multiplication
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2.2 Ordering Properties of the Real
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(c) For all real numbers a, a2 ≥
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2.3 Absolute Value 55 (⇐) Now sup
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2.4 The Division Algorithm 57 mn =
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2.5 Divisibility and Prime Numbers
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2.5 Divisibility and Prime Numbers
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Sets and Their Properties 3 All of
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U A B Figure 3.5 Venn diagram illus
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(f) For all sets A, B, and C,ifA
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3.2 Proving Basic Set Properties 69
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3.3 Families of Sets 71 Notice that
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Example 3.3.2 Consider the family o
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3.3 Families of Sets 75 x ∈ � A
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3.3 Families of Sets 77 and ... and
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Example 3.4.2 For any positive inte
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3.4 The Principle of Mathematical I
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Proof. To clean up the notation, we
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3.5 Variations of the PMI 85 EXERCI
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(g) (a −m ) −n = a mn (h) (ab)
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Strong Induction 3.5 Variations of
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3.6 Equivalence Relations 91 EXERCI
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3.6 Equivalence Relations 93 beginn
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3.6 Equivalence Relations 95 constr
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3.7 Equivalence Classes and Partiti
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3.8 Building the Rational Numbers 1
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3.8 Building the Rational Numbers 1
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3.10 Irrational Numbers 107 EXERCIS
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3.10 Irrational Numbers 109 To the
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EXERCISE 3.10.2 √ 2 is irrational
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(a) {(x, y) : x ≤ y} (b) {(x, y)
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3.11 Relations in General 115 (O3)
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3.11 Relations in General 117 at al
Page 142 and 143: Functions 4 Second only to sets, fu
Page 144 and 145: 4.1 Definition and Examples 121 pro
Page 146 and 147: Solution We show that T satisfies p
Page 148 and 149: 4.2 One-to-one and Onto Functions 4
Page 150 and 151: 4.2 One-to-one and Onto Functions 1
Page 152 and 153: A 1 x A B f (A 1 ) Figure 4.4 The i
Page 154 and 155: 4.4 Composition and Inverse Functio
Page 156 and 157: 4.4 Composition and Inverse Functio
Page 158 and 159: 4.5 Three Helpful Theorems 135 The
Page 160 and 161: 4.6 Finite Sets 137 EXERCISE 4.5.3
Page 162 and 163: 4.7 Infinite Sets 139 The next exer
Page 164 and 165: 4.7 Infinite Sets 141 of A. This or
Page 166 and 167: 4.7 Infinite Sets 143 EXERCISE 4.7.
Page 168 and 169: EXERCISE 4.8.3 If A1,A2, and B are
Page 170 and 171: 4.8 Cartesian Products and Cardinal
Page 172 and 173: 4.8 Cartesian Products and Cardinal
Page 174 and 175: 4.9 Combinations and Partitions 151
Page 180 and 181: 4.10 The Binomial Theorem 157 EXERC
Page 182 and 183: EXERCISE 4.10.3 Determine the follo
Page 184 and 185: (c) The coefficient of ab 3 c 2 in
Page 186 and 187: P A R T I I Basic Principles of Ana
Page 188 and 189: The Real Numbers 5 Let’s look at
Page 190 and 191: 5.1 The Least Upper Bound Axiom 167
Page 194 and 195: 5.2 The Archimedean Property 171 No
Page 196 and 197: 5.3 Open and Closed Sets 173 Thus A
Page 198 and 199: 5.4 Interior, Exterior, Boundary, a
Page 200 and 201: 5.4 Interior, Exterior, Boundary, a
Page 202 and 203: 5.5 Closure of Sets 179 The constru
Page 204 and 205: 5.6 Compactness 181 EXERCISE 5.6.3
Page 206 and 207: 5.6 Compactness 183 EXERCISE 5.6.13
Page 208 and 209: Sequences of Real Numbers 6.1 Seque
Page 210 and 211: 6.1 Sequences Defined 187 Example 6
Page 212 and 213: EXERCISE 6.1.15 Consider the sequen
Page 214 and 215: L 1 e L L 2 e . . . . . . 1 2 3 4 N
Page 216 and 217: 6.2 Convergence of Sequences 193 EX
Page 218 and 219: 6.2 Convergence of Sequences 195 To
Page 220 and 221: 6.3 The Nested Interval Property 19
Page 226 and 227: a1 a4 aN 1 2 aN aN 1 1 aN 1 3 a5 a3
Page 228 and 229: Example 6.4.7 The terms a0 = 1 a1 =
Page 230 and 231: Functions of a Real Variable 7 Func
Page 232 and 233: 7.1 Bounded and Monotone Functions
Page 234 and 235: 50 40 30 20 7 1 ε 7 7 2 ε 0 ( ( F
Page 236 and 237: 7.2 Limits and Their Basic Properti
Page 238 and 239: 7.2 Limits and Their Basic Properti
Page 240 and 241: 7.3 More on Limits 217 values of 0
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7.4 Limits Involving Infinity 219 W
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7.4 Limits Involving Infinity 221 T
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(a) limx→a f(x) =−∞ (b) limx
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7.5 Continuity 225 If f is not cont
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1 2 3 4 5 6 Figure 7.6 Some example
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|a − x| |xa| = |x − a| |x||a| 7
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7.6 Implications of Continuity 231
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7.6 Implications of Continuity 233
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7.7 Uniform Continuity 235 we decla
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7.7 Uniform Continuity 237 Next, le
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7.7.2 Uniform Continuity and Compac
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P A R T I I I Basic Principles of A
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Groups 8 In its simplest terms, alg
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8.1 Introduction to Groups 245 are
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8.1 Introduction to Groups 247 Defi
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◦ 0 1 2 3 4 5 0 0 1 2 3 4 5 1 1 0
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Show that C × is an abelian group
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8.2 Subgroups 253 G1 and G3 are aut
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8.2 Subgroups 255 Notice how proper
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8.2 Subgroups 257 Example 8.2.15 su
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EXERCISE 8.2.23 If G is a group and
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8.3 Quotient Groups 261 [0] ={...,
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8.3 Quotient Groups 263 is standard
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Step 2: Define an operation ∗H on
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(iii) (Z + 3.8) +Z (Z + 1.2) (iv)
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then for a given f ∈ S and any a
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Thus f 2 = f ◦ f = (132)(56) ◦
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2 3 1 3 1 Figure 8.1 Rigid square u
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EXERCISE 8.4.12 Complete Table 8.64
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8.5 Normal Subgroups 277 Theorem 8.
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8.5 Normal Subgroups 279 point to b
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8.6 Group Morphisms 281 EXERCISE 8.
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8.6 Group Morphisms 283 Notice that
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Ker � e x 2 x maps to x Ker � G
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Rings 9 We can create algebraic str
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9.1 Rings and Fields 289 (R9) Multi
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9.1 Rings and Fields 291 we define
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9.2 Subrings 293 In addition to pro
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9.2 Subrings 295 p1 = q1 = 3. Verif
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9.3 Ring Properties 297 Theorem 9.3
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9.3 Ring Properties 299 EXERCISE 9.
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9.4 Ring Extensions 301 in the inte
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9.4 Ring Extensions 303 Example 9.4
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9.4 Ring Extensions 305 We insist t
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9.5 Ideals 307 An ideal, say a left
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9.6 Generated Ideals 309 Proof. Sup
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EXERCISE 9.6.6 In the integers, wha
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9.7 Prime and Maximal Ideals 313 Ev
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9.8 Integral Domains 315 EXERCISE 9
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9.8 Integral Domains 317 Corollary
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9.9 Unique Factorization Domains 31
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9.10 Principal Ideal Domains 321 al
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9.10 Principal Ideal Domains 323 So
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9.11 Euclidean Domains 325 next exe
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9.11 Euclidean Domains 327 Exercise
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Proof. Let f and g be nonzero polyn
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9.12 Polynomials over a Field 331 W
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9.13 Polynomials over the Integers
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9.14 Ring Morphisms 335 Example 9.1
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9.14 Ring Morphisms 337 EXERCISE 9.
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9.15 Quotient Rings 339 this, howev
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9.15 Quotient Rings 341 Because the
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9.15 Quotient Rings 343 However, th
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Index =,51 ɛ (epsilon), 167-168 ɛ
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Index 347 Cosets, 263, 264, 265 Lag
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Index 349 Gauchy sequences, 202-206
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Index 351 identity, 124 relations v
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Index 353 Quaternions, 248, 253, 25
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Index 355 Tower of Hanoi, 84 Transc
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