Foundations Of Mathematical Analysis Johnsonbaugh Solutions Pdf

Errata

I

Sets and Functions

In this brief chapter we will summarize some of the fundamental notation and definitions which will be used throughout the text.

1. Sets

As a starting point we describe what we mean by a set. We will not attempt to define the term set, but we demonstrate the use of the term by examples. The term set is used (roughly) to describe any collection of objects. For example, the set whose objects are the positive integers 1, 2, 3 may be denoted

{1, 2, 3}

If a set consists of a finite number of objects, we may denote the set by listing its objects. If a set consists of infinitely many objects such a listing is impossible. In this case we may describe the set by naming a property common to all the objects in the set. For example, to describe the set of positive integers, we use the notation

{x x is a positive integer}

This notation is read "the set of all x such that x is read such that.") In general, to describe the set of all objects having a particular property P, we write

{x x has property P}

If x is an object in a set A, we write

x ∈ A

and say that x is an element (or a point) of A. If x is not an element of the set A, we write

x A

We shall say two sets A and B are equal and write A = B if A and B have the same elements. Thus A = B if whenever x ∈ A, then x ∈ B, and whenever x ∈ B, then x ∈ A.

Definition 1.1 If A and B are sets, the union of A and B is the set

A ∪ B = {x x ∈ A or x ∈ B}

The intersection of A and B is the set

A ∩ B = {x x ∈ A or x ∈ B}

For example, if A = {1, 2, 3} and B = {2, 3, 4}, then

A ∪ B = {1, 2, 3, 4}

A ∩ B = {2, 3}

is a collection of sets, we define

∪ = {x x ∈ A for some A ∈ }

∩ = {x x ∈ A for some A ∈ }

= {A1, A2, A3, . . .}, we write

For example, if

= {A1, A2,…}

whereA1 = {1}, A2 = {1, 2},…,An = {1, 2,…, n},…

is the set of positive integers.

Definition 1.2 The empty set is the set with no elements and is denoted ø.

Definition 1.3 If every element of the set A is an element of the set B, we write A ⊂ B and say that A is contained in B or that A is a subset of B.

We write B ⊃ A if A ⊂ B and say that B contains A.

We note that according to Definition 1.3, A ⊂ B allows the possibility that A = B. It follows immediately from our definitions that A = B if and only if A ⊂ B and B ⊂ A. We call A a proper subset of B if A ⊂ B and A ≠ B.

Definition 1.4 If A and B are sets, the difference of A and B is the set

A\B = {x x ∈ A or x B}

For example, if A = {1, 2, 3} and B = {2, 3, 4}, then A\B = {1}.

If we are working with sets all of which are subsets of some particular set U, we sometimes say that U is the universe in which we are working. For example, if we are working with sets of integers, we could agree that the universe U is the set Z of all integers.

Definition 1.5 If we are working in a fixed universe U, and A ⊂ U, we write

A′ = U\A

The set U\A is called the complement of A relative to U (or just simply the complement of A if the universe U is understood).

For example, if U is the set of real numbers, we would state that the complement of the set of rational numbers is the set of irrational numbers.

We now prove two theorems which are known as De Morgan's laws. These theorems can often be used to convert a statement about unions into a statement about intersections and vice versa.

Theorem 1.6 If A and B are subsets of a universe U, then

(A ∪ B)′ = A′ ∩ B′ (A ∩ B)′ = A′ ∪ B′

Proof. We prove only the first equation leaving the second as an exercise.

Let x ∈ (A ∪ B)′. Then x (A ∪ B), and hence x A and x B. Thus x ∈ A′ and x ∈ B′ which implies x ∈ A′ ∩ B′. Therefore,

(A ∪ B)′ = A′ ∩ B′

Next suppose x ∈ A′ ∩ B′. Then x ∈ A′ and x ∈ B′, and hence x A and x B. Thus x A ∪ B, which implies x ∈ (A ∪ B)′. Therefore,

A′ ∩ B′ ⊂ (A ∪ B)′

In a similar way one may prove the following theorem.

Theorem 1.7 be a collection of subsets of a universe U, and let

′ \ A }

Exercises

1.1 Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {1, 2, 3, 4}

B = {1, 3, 5, 7, 9}

Compute the following sets:

(a) A ∪ B

(b) A ∩ B

(d) B\A

(e) A′

(f) B′

1.2 Prove the second equation of Theorem 1.6.

1.3 Prove Theorem 1.7.

In exercises 4 to 13, A, B, and C are all subsets of a universe U.

1.4 Prove: A ⊂ ø if and only if A = ø

1.5 Prove: If A ⊂ B and B ⊂ C, then A ⊂ C

1.6 Prove: A ∪ B = B ∪ A

1.7 Prove: A ∪ (B ∪ C) = (A ∪ B) ∪ C

1.8 Prove: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

1.9 Prove: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∩ C)

1.10 Prove: (A′)′ = A

1.11 Prove: If A ⊂ B, then B′ ⊂ A′

1.12 Prove: A ∩ A′ = ø

1.13 Prove: A B if and only if A ∩ B means is not a subset of)

1.14 Let U be a universe. Let A be a subset of U be a collection of subsets of U. Prove:

2. Functions

The concept of a function is central to all of mathematics, and in this section we will give a precise definition of a function and prove several properties of functions. We begin with the concept of an ordered pair.

Definition 2.1 The ordered pair of elements a and b, written (a, b), is the set

(a,b) = {{a}, {a, b}}

a is called the first element of (a, b) and b is called the second element of (a, b).

The crucial property of an ordered pair is stated in the next theorem. The theorem states that two ordered pairs are identical if and only if they have the same first elements and the same second elements.

Theorem 2.2 Let (a, b) and (c, d) be ordered pairs. Then (a, b) = (c, d) if and only if a = c and b = d.

Proof. Suppose a = c and b = d. Then {a} = {c} and {a, b} = {c, d}, and therefore (a, b) = (c, d).

Conversely, suppose (a, b) = (c, d). Then

{{a}, {a, b}} = {{c}, {c, d}}(2.1)

First, we consider the case that a = b. Now

a,b = {{a}, {a, b}} = {{a}, {a}} = {{a}}

and so {{c}, {c, d}} = {{a}}

Thus {a} = {c} = {c, d}, and therefore a = b = c = d.

Now suppose a ≠ b. From equation (2.1) we see that either {c} = {a} or {c} = {a, b}. Since a ≠ b, we must have {c} = {a}, which implies that a = c. Again using equation (2.1), we have that either {a, b} = {c} or {a, b} = {c, d}. If {a, b} = {c}, then a = b = c, which is not the case, and thus {a, b} = {c, d}. It follows that b = c or b = d, but if b = c, we would have the contradiction b = c = a. Therefore b = d

Definition 2.3 If X and Y are sets, the Cartesian product of X and Y denoted X × Y is the set

X × Y = {(x, y) x ∈ X and y ∈ Y}

Definition 2.4 Let X and Y be sets. A function from X into Y is a subset f of X × Y satisfying

(i) If (x, y) and (x, y′) belong to f, then y = y′.

(ii) If x ∈ X, then (x, y) ∈ f for some y ∈ Y.

If f is a function from X into Y, we will write f: X → Y.

The crucial property of a function from X into Y is that with each [2.4(ii)] element x in X there is associated a unique [2.4(i)] element y in Y.

Definition 2.5 Let f be a function from X into Y. Let A ⊂ X and B ⊂ Y .

(i) X is called the domain of f. Y is called the codomain of f.

(ii) If (x, y) ∈ f, we write y = f(x) and call y the (direct) image of x under f.

(iii) The range of f is the set

{f(x) x ∈ X}

(iv) The image of A under f is the set f(A) = {f(x) x ∈ A}.

(v) The inverse image of B under f is the set

f−1(B) = {x f(x) ∈ B}

(vi) f is onto Y if f(X) = Y.

(vii) f is one to one if f(x) = f(x′) always implies that x = x′ for x, x′ ∈ X.

If f is a one-to-one function from X into Y, we may define the inverse function to f, denoted f −1, from the range of f onto X by the rule

(y, x) ∈ f −1 if and only if (x, y) ∈ f

Notice that the function f −1 is defined only if f is one to one but that f −1(B) is defined for an arbitrary function f and for all sets B ⊂ Y.

If g: X → Y and f: Y → Z, we define the composition f ∘ g: X → Z by the rule

(f ∘ g)(x) = f(g(x)) for each x ∈ X

If f: X → Y and A ⊂ X, we define the function

f A = {(x, y) ∈ f x ∈ A}

f A is called the restriction of f to A, and f is said to an extension of f A to X.

We illustrate the preceding definitions with an example.

Example. Let

f = {(1, 1), (2, 1), (3, 4)}, A = {(1, 2)}, B = {1}

The domain of f is {1, 2, 3} and the range of f is {1, 4}. The image of A under f is the set f(A) = {1}. The inverse image of B under f is the set f −1(B) = {1, 2}. If we let Y = {1, 4}, f is onto Y. The function f is not one to one. The restriction of f to A is the function

f A = {(1, 1), (2, 1)}

Letg = {(1, 1), (2, 3)}

Then g is one to one and the inverse function to g is the function

g−1 = {(1, 1), (3, 2)}

The composition f ∘ g is the function

f ∘ g = {(1, 1), (2, 4)}

The next theorem establishes several properties of inverse and direct images.

Theorem 2.6 Let f be a function from X into Y be a collection of subsets of X be a collection of subsets of Y. Let C ⊂ Y.

Then

(iv) f −1(C′) = [f −1(C)]′

Proof. We prove part (ii) only. Let x ). Then f(x) , and thus f(x) ∈ C1 for some C . Thus x ∈ f−1(C1), and therefore x ∈ ∪ {f−1(C): C }. We have proved that

Now let x ∈ ∪ {f−1(C): C }, then x ∈ f−1(C1) for some C . Therefore f(x) ∈ C1, and hence f(x) . It follows that x ∈ f ). We have proved that

= {A, B = {C, D} the conclusions of Theorem 2.6 may be written

It is not true that in general one has f(A ∩ B) = f(A) ∩ f(B). (Verify.)

Exercises

2.1 Let g = {(1, 2), (2, 2), (3, 1), (4, 4)}

f = {(1, 5), (2, 7), (3, 9), (4, 17)}

A = {1, 2}

Compute

(a) The domain of g

(b) The range of g

(d) g−1(A)

(e) f ∘ g

(f) f −1

2.2 Let h: X → Y, g: Y → Z, and f: Z → W. Prove that (f ∘ g) ∘ h = f ∘ (g ∘ h).

2.3 Let f: X → Y and A ⊂ X and B ⊂ Y. Prove

(a) f(f −1(B)) ⊂ B

(b) A ⊂ f −1(f(B))

2.4 Prove Theorem 2.6 (i), (iii), and (iv).

2.5 Let g: X → Y and f: Y → Z.

(a) Prove that if g and f are one-to-one functions, then f ∘ g is a one-to-one function and (f ∘ g)−1 = g−1 ∘ f −1.

(b) Prove that if g and f are onto functions, then f ∘ g is an onto function.

2.6 Let f: X → Y. Prove that f is a one-to-one function if and only if

f(A ∩ B) = f(A) ∩ f(B)

for all subsets A and B of X.

II

The Real Number System

The central topic of study in this text is the real number system. We will define the real numbers by specifying which axioms or rules the real numbers are assumed to satisfy. In an appropriate theory of sets, one may construct a number system which satisfies these axioms [see Kelley (1955) and Landau (1960)]. It can be shown that these axioms determine the real numbers (see Exercise 7.9).

3. The Algebraic Axioms of the Real Numbers

We begin with a definition.

Definition 3.1 A binary operation on a set X is a function from X × X into X.

Intuitively, a binary operation on a set X is a rule which associates with each ordered pair of elements of X a unique element of X. Binary operations are often written +, ·, or *, and the value of the function at an ordered pair (x, y) is usually written x + y, x · y, or x*y. The particular binary operations with which we shall be immediately concerned are addition and multiplication of real numbers. Addition or multiplication of real numbers associates with each ordered pair of real numbers (a, b) another real number, namely the sum of a and b or the product of a and b.

Definition 3.2 The real numbers R is a set of objects satisfying Axioms 1 to 13 as listed in the following:

Axiom 1. There is a binary operation called addition and denoted + such that if x and y are real numbers, x + y is a real number.

Axiom 2. Addition is associative.

(x + Y) + Z = x + (y + z)

for all x, y, z ∈ R.

Axiom 3. Addition is commutative.

x + y = y + x

for all x, y ∈ R.

Axiom 4. An additive identity exists. There exists a real number denoted 0 which satisfies

x + 0 = x = 0 + x

for all x ∈ R.

Axiom 5. Additive inverses exist. For each x ∈ R, there exists y ∈ Rsuch that

x + y = 0 = y + x

The number y of Axiom 5 may be shown to be unique, and it is denoted −x. We define x − z as x + (−z) for all x and z in R.

Any mathematical system which satisfies axioms 1 to 5 is called a commutative (or abelian) group. Using axioms 1 to 5 one may establish the usual rules for addition of real numbers. We give one example in the next theorem and several other additive properties are given as exercises.

Theorem 3.3 The additive identity of axiom 4 is unique, that is, if there exists 0′ ∈ R such that x + 0′ = x for all x ∈ R, then 0 = 0′.

Proof. Suppose there exists 0′ ∈ R such that x + 0′ = x for all x ∈ R. Then 0 + 0′ = 0. On the other hand, from axiom 4 we have 0′ + 0 = 0′. Since addition is commutative (axiom 3),

0 = 0 + 0′ = 0′ + 0 = 0′

We continue by giving the axioms for multiplication of real numbers.

Axiom 6. There is a binary operation called multiplication and denoted such that if x and y are real numbers, then x · y (or xy) is a real number.

Axiom 7. Multiplication is associative.

(xy)z = x(yz)

for all x, y, z ∈ R.

Axiom 8. Multiplication is cummutative.

xy = yx

for all x, y ∈ R.

Axiom 9. A multiplicative identity exists. There exists a real number, different from 0, denoted 1 which satisfies

x·1 = x = 1·x

for all x in R.

Axiom 10. Multiplicative inverses exist for nonzero real numbers. For any x ∈ R with x ≠ 0, there exists y ∈ R such that

xy = 1 = yx

The next axiom links the operations of addition and multiplication.

Axiom 11. Multiplication distributes over addition.

x(y + z) = xy + xz (y + z)x = yx + zx

for all x, y, z ∈ R.

As in the case of addition one may show that 1 and multiplicative inverses are unique. The multiplicative inverse of a nonzero real number x is denoted x−1 or 1/x. We define x/y = x(y−1) if y ≠ 0. Any mathematical system satisfying axioms 1 to 11 is called a field. Using axioms 1 to 11 one may establish all of the well-known algebraic properties of the real numbers. We give one example; others are given in the exercises. In subsequent sections we will assume that all of the well-known algebraic properties of the real numbers have been verified from the axioms. The interested reader may consult Landau (1960) to see how this might be accomplished.

Theorem 3.4 x · 0 = 0 for all x in R.

Proof. x·(0 + 0) = x·0 by axiom 4. On the other hand x·(0 + 0) = x·0 + x·0 by axiom 11. Thus x·0 = x·0 + x·0. Now

x·0 + [–(x·0)] = (x·0 + x·0) + [–(x·0)]

Again using axiom 5 and axiom 2, we have

0 = x·0 + 0

and so by axiom 4 we have 0 = x

Exercises

3.1 Prove that the additive inverse of axiom 5 is unique.

3.2 Prove that

(a) (x + y) + (z + w) = (x + (y + z)) + w for all x, y, z, w ∈ R

(b) The sum x + y + z + w is independent of the manner in which the parentheses are inserted.

3.3 Prove that −(−x) = x for all x ∈ R.

3.4 Prove that −(x + y) = −x − y for all x, y ∈ R.

3.5 Let x, y ∈ R. Prove that xy = 0 if and only if x = 0 or y = 0.

3.6 Let x, y ∈ R. Prove that if xy = xz and x ≠ 0, then y = z.

3.7 Prove that −(xy) = x(−y) = (−x)y for all x, y ∈ R.

3.8 Prove that (−1)x = −x for all x ∈ R.

4. The Order Axiom of the Real Numbers

In this section we give the order axiom of the real numbers and derive several useful results.

Axiom 12. There is a subset P of R called the positive real numbers satisfying

(i) If x and y are in P, then x + y and xy are in P.

(ii) If x is in R, exactly one of the following statements is true:

x ∈ P or x = 0 or –x ∈ P

Using axiom 12 we can define the usual notation for order.

Definition 4.1 Let x and y be real numbers.

(i) x is negative if −x is positive.

(ii) x > y means x − y is positive.

(iii) x ≥ y means x > y or x = y.

(iv) x < y means y > x.

(v) x ≤ y means y ≥ x.

The inequality x > y (x < y) is read "x is greater (less) than y" and the inequality x ≥ y (x ≤ y) is read "x is greater (less) than or equal to y." Several order properties of the real numbers are given in the next theorem.

Theorem 4.2

(i) 1 > 0.

(ii) If x > y and y > z, then x > z, x, y, z ∈ R.

(iii) If x > y, then x + z > y + z, x, y, z ∈ R.

(iv) If x > y and z > 0, then xz > yz, x, y, z ∈ R.

(v) If x > y and z < 0, then xz < yz, x, y, z ∈ R.

Proof. We prove parts (i) and (ii) leaving the others as exercises. By axiom 12, exactly one of the following statements is true

1 ∈ P or 1 = 0 or –1 ∈ P

By axiom 9, 1 ≠ 0. Suppose – ∈ P. Then by axiom 12, (–1)(–1) = 1 ∈ P(see Exercise 3.8) and hence 1 ∈ P and –1 ∈ P which contradicts axiom 12. Therefore 1 ∈ P and (i) holds.

If x > y and y > z, then x – y, y – z ∈ P by Definition 4.1. By axiom 12, x – z = (x – y) + (y – z) ∈ P and thus x > z

We will employ the following notation throughout the remainder of the text.

Definition 4.3 Let a, b ∈ R with a < b. We define

We call (c, d) an open interval; [c, d] a closed interval; and either [c, d) or (c, d] a half-open interval, where c or d are possibly ± ∞.

We conclude this section by defining the absolute value of a real number and deriving several results concerning absolute value.

Definition 4.4 Let x be a real number. We define

x the absolute value of x.

Theorem 4.5

(i) Let ε x < ε if and only if −ε < x < ε x ≤ ε if and only if − ε ≤ x ≤ ε.

(ii) x ≤ x for all x ∈ R.

xy x y for all x, y ∈ R.

x + y x y for all x, y ∈ R.

Proof. We prove only (ii) and (iv) leaving the others as exercises.

If x ≥ 0, then x x x . If x < 0, then x x x . In either case x x .

If x + y x + y = x + y ≤ x + y by (ii). If x + y x + y = −(x + y) = −x − y ≤ x + y

Exercises

4.1 Prove parts (iii), (iv), and (v) of Theorem 4.2.

4.2 Prove parts (i) and (iii) of Theorem 4.5.

4.3 Prove that if x ≤ y and y ≤ x, then x = y, x, y ∈ R.

4.4 Prove that if xy > 0, then either x > 0 and y > 0 or x < 0 and y < 0, x, y ∈ R.

4.5 Prove that if x > 0, then 1/x >, x ∈ R.

4.6 Prove that x² > 0 for all x ∈ R with x ≠ 0.

4.7 Prove that x² + y² ≥ 2xy for all x, y ∈ R.

4.8 Prove that if 0 < xy and x < y, then 1/y < 1/x, x, y ∈ R.

4.9 Prove that if x ≤ y + ε for every ε > 0, then x ≤ y, x, y ∈ R.

4.10 Prove that if x + y > z and w > x, then w + y > z, x, y, z, w ∈ R.

5. The Least-Upper-Bound Axiom

Before giving the final axiom for the real numbers we must make some definitions.

Definition 5.1 A nonempty subset X of R is said to be bounded above (below) if there exists a real number a such that x ≤ a (x ≥ a) for all x ∈ X. The number a is called an upper (lower) bound for X.

Definition 5.2 Let X be a nonempty subset of R.

A number a in R is said to be a least upper bound for X if

(i) a is an upper bound for X.

(ii) If b is an upper bound for X, then a ≤ b.

A number a in R is said to be a greatest lower bound for X if

(i) a is a lower bound for X.

(ii) If b is a lower bound for X, then b ≤ a.

Part (ii) of Definition 5.2 concerning least upper bounds may be equivalently stated as follows: if b < a, then b is not an upper bound for X. By Definition 5.1, this last statement is equivalent to

(ii′) If b < a, there exists x ∈ X such that b < x.

In proving that a is a least upper bound for a set X, it is sometimes easier to use (ii′) rather than (ii). Part (ii) of Definition 5.2 concerning greatest lower bounds may be rephrased in a similar way.

The next theorem states that if a set has a least upper bound, it is unique.

Theorem 5.3 Let X be a subset of R. If a and b are least upper bounds for X, then a = b.

Proof. By Definition 5.2, since a is a least upper bound for X and b is an upper bound for X, we have a ≤ b. Similarly, b ≤ a, and thus a = b

Theorem 5.3 (and the corresponding result for greatest lower bounds) allows us to speak of the least upper bound (or the greatest lower bound) and justifies the following notation. If a is the least upper bound of a set X, we let a = lub X. Similarly, if b is the greatest lower bound of a set X, we let b = glb X. If X is finite set, we also denote lub X by max X, and we denote glb X by min X.

For example,

lub(0, 1) = 1 = lub[0,1]

and glb(0, 1) = 0 = glb[0,1]

The least upper bound of a set X is also called the supremum of X, and the greatest lower bound of a set X is also called the infimum of X. The notation lub X = sup X and glb X = inf X is also used.

Axiom 13. A nonempty subset of real numbers which is bounded above has a least upper bound.

Figure 5.1

Axiom 13 is called the completeness axiom for the real numbers. The real numbers are complete in the sense that there are no holes in the real line. Informally, if there were a hole in the real line (see Figure 5.1), the set of numbers to the left of the hole would have no least upper bound.

The least-upper-bound axiom is the basis of many deep theorems in analysis concerning the real numbers. Many theorems about the real numbers (for example, Theorems 16.2 and 18.1) which involve the existence of a number with special properties ultimately rest on this axiom.

Using axiom 13, we are able to prove the existence of greatest lower bounds.

Theorem 5.4 A nonempty subset of real numbers which is bounded below has a greatest lower bound.

Proof. Let X be a nonempty subset of real numbers which is bounded below, and let Y be the set of lower bounds for X. Let c ∈ X. Then y ≤ c for y ∈ Y. Thus Y is bounded above, and by the least-upper-bound axiom Y has a least upper bound a. We will show that a is the greatest lower bound of X.

Let x ∈ X. Then y ≤ x for all y ∈ Y, and thus x is an upper bound for Y. Since a is the least upper bound of Y, we have a ≤ x. Therefore, a is a lower bound for X.

Let b any lower bound for X. Then b ∈ Y, and hence b ≤ a. By Definition 5.2, a is the greatest lower bound of X

Exercises

5.1 Let X be a set of real numbers with least upper bound a. Prove that if ε > 0, there exists x ∈ X such that a − ε < x ≤ a.

5.2 Prove that the greatest lower bound of a set of real numbers is unique.

5.3 Give another proof of Theorem 5.4 by considering the least upper bound of the set

−X = {−x x ∈ X}

5.4 Show that if X is a nonempty subset of real numbers which is bounded above then

lub X = −glb(−X)

5.5 Let X and Y be nonempty subsets of real numbers such that X ⊂ Y and Y is bounded above. Prove that

lub X ≤ lub Y

5.6 Let X be a set of real numbers with least upper bound a. Let t ≥ 0. Prove that ta is the least upper bound of the set

tX = {tx x ∈ X}

State and prove an analogous result if t < 0.

5.7 Let X and Y be sets of real numbers with least upper bounds a and b, respectively. Prove that a + b is the least upper bound of the set

X + Y = {x + y x ∈ X, y ∈ Y}

5.8 * If f is a real-valued function on (a, b) and c ∈ (a, b), we say that f is strictly increasing at c if there exists δ > 0 such that if c − δ < x < c, then f(x) < f(c) and if c < x < c + δ, then f(c) < f(x). We say that f is strictly increasing on (a, b) if whenever x, y ∈ (a, b) with x < y, we have f(x) < f(y). Prove that if f is strictly increasing at each point of (a, b), then f is strictly increasing on (a, b).

6. The Set of Positive Integers

We now isolate that subset of the real numbers known as the positive integers. We will define the positive integers as the smallest subset P of R having the properties that 1 ∈ P and if n ∈ P, then n + 1 ∈ P.

Definition 6.1 A subset X of R is said to be a successor set

(i) If 1 ∈ X,

(ii) If n ∈ X, then n +