Elementary Real and Complex Analysis

By Georgi E. Shilov

In this book the renowned Russian mathematician Georgi E. Shilov brings his unique perspective to real and complex analysis, an area of perennial interest in mathematics. Although there are many books available on the topic, the present work is specially designed for undergraduates in mathematics, science and engineering. A high level of mathematical sophistication is not required.
The book begins with a systematic study of real numbers, understood to be a set of objects satisfying certain definite axioms. The concepts of a mathematical structure and an isomorphism are introduced in Chapter 2, after a brief digression on set theory, and a proof of the uniqueness of the structure of real numbers is given as an illustration. Two other structures are then introduced, namely n-dimensional space and the field of complex numbers.
After a detailed treatment of metric spaces in Chapter 3, a general theory of limits is developed in Chapter 4. Chapter 5 treats some theorems on continuous numerical functions on the real line, and then considers the use of functional equations to introduce the logarithm and the trigonometric functions. Chapter 6 is on infinite series, dealing not only with numerical series but also with series whose terms are vectors and functions (including power series). Chapters 7 and 8 treat differential calculus proper, with Taylor's series leading to a natural extension of real analysis into the complex domain. Chapter 9 presents the general theory of Riemann integration, together with a number of its applications. Analytic functions are covered in Chapter 10, while Chapter 11 is devoted to improper integrals, and makes full use of the technique of analytic functions.
Each chapter includes a set of problems, with selected hints and answers at the end of the book. A wealth of examples and applications can be found throughout the text. Over 340 theorems are fully proved.

• Language: English
• Publisher: Dover Publications
• Release date: Jul 31, 2012
• ISBN: 9780486135007

Book preview

G.E.S.

1 Real Numbers

1.1. Set-Theoretic Preliminaries

1.11. Words like aggregate, collection, and set come up at once when talking about objects (or elements) of any kind. Thus one can talk about the set of students in an auditorium, the set of grains of sand on a beach, the set of vertices or the set of sides of a polygon, and so on. In each of these examples the set in question consists of a definite number of elements, which can be estimated within certain limits, even though it may be difficult in practice to find the number of elements exactly.† Such sets are said to be finite.

In mathematics one must often deal with sets consisting of a number of objects which is not finite. The simplest examples of such sets are the set 1, 2, 3, ... of all natural numbers (positive integers) and the set of all the points on a line segment (precise definitions of these objects will be given later). Such sets are said to be infinite. To the category of sets we also assign the empty set, namely, the set containing no elements of all.

As a rule, sets will be denoted by large letters A, B, C, ... and elements of sets will be denoted by small letters. By a A (or A a) we mean that a is an element of the set A, while by a A (or a A) we mean that a is not an element of the set A. By A B (or B A) we mean that every element of the set A is an element of the set B; the set A is then said to be a subset of the set B. The largest subset of the set B is obviously the set B itself, while the smallest subset of B is the empty set. Any other subset of the set B, containing some but not all elements of B, is called a proper subset of Bare called inclusion relations. Suppose both A B and B A. Then every element of the set A is an element of the set B, and conversely, every element of the set B is an element of the set A. It follows that the sets A and B consist of precisely the same elements and hence coincide, a fact expressed by writing A = B. The analogous formula for elements, namely a = b, simply means that a and b are one and the same element.

Sets can be specified in various ways. The simplest way is to write the elements of the set explicitly between curly brackets; e.g., A = {1, 2, 3, ...} is the set of all natural numbers (positive integers). Another way is to specify some property of the elements of the set; e.g., A = {x: x0} is the set of all x satisfying the inequality x0 written after the colon.

1.12. Unions and intersections. Let A, B, C, ... be given sets. Then the set of all elements of A, B, C, ... belonging to at least one of the sets A, B, C, ... is called the union of the sets A, B, C, ..., while the set of all elements of A, B, C, ... belonging to every one of the sets A, B, C, ... is called the intersection of the sets A, B, C, .... For example, let

A = {6, 7, 8, ...}, B = {3, 6, 9, ...},

i.e., let A be the set of all natural numbers greater than 5 and B the set of all natural numbers divisible by 3. Then the union of A and B is the set

S = {3, 6, 7, 8, 9, 10, ...},

the set of all natural numbers except 1, 2, 4, and 5, while the intersection of A and B is the set

D = {6, 9, 12 ...},

the set of all natural numbers divisible by 3 except 3 itself. In the case where the sets A, B, C, ... have no elements in common, the intersection of A, B, C, ... is the empty set, and the sets A, B, C, ... are then said to be nonintersecting. For example, the three sets

A = {1, 2},B = {2, 3},C = {1, 3}

are nonintersecting, even though any two of the sets share a common element.

We can consider the union and intersection of both a finite number of sets and an infinite number of sets. For example, the union of the sets of points on all (infinitely many) lines in the plane passing through a given point O is clearly the set of all points in the plane, while the intersection of all these sets consists of the single point O.

To denote the union S of given sets, say A1, A2, ..., Av, ..., we use the symbols

,

writing

while to denote the intersection D of the sets, we use the symbols

,

writing

1.2. Axioms for the Real Number System

The following considerations stem from the simplest properties of numbers, known partly from everyday experience and partly from elementary mathematics.† Rather than define the real numbers separately, we will define the whole set of real numbers at once as a set of elements equipped with certain operations and relations which in turn satisfy four groups of axioms. The first group consists of the addition axioms, the second of the multiplication axioms, the third of the order axioms, and the fourth of a single axiom called the least upper bound axiom.

Definition. By the real number system R is meant the set whose elements x, y, z, ..., called real numbers, satisfy the four groups of axioms given in Secs. 1.21–1.24. The set R is often called the real line, with its elements in turn called points.

1.21. The addition axioms. To every pair of elements x and y in R there corresponds a (unique) element x + y, called the sum of x and y, where the rule associating x + y with x and y has the following properties:

(a) x + y = y + x for every x and y in R (addition is commutative);

(b) (x + y) + z = x + (y + z) for every x, y, z in R (addition is associative);‡

(c) R contains an element 0, called the zero element, such that x + 0 = x for every x in R;

(d) For every x in R there exists an element y in R, called the negative of x, such that x + y = 0.

1.22. The multiplication axioms. To every pair of elements x and y in R there corresponds a (unique) element x · y (or xy), called the product of x and y, where the rule associating xy with x and y has the following properties:

(a) xy = yx for every x and y in R (multiplication is commutative);

(b) (xy)z = x(yz) for every x, y, z in R (multiplication is associative);†

(c) R contains an element 1 ≠ 0, called the unit element, such that 1 · x = x for every x in R;

(d) For every x ≠ 0 in R there exists an element u in R, called the reciprocal of x, such that xu = 1;

(e) The formula

x(y + z) = xy + xz

holds for every x, y, z in R (multiplication is distributive over addition).

The last axiom connects the operation of multiplication with the operation of addition introduced in Sec. 1.21.

A set of objects x, y, z, ... satisfying the axioms of Secs. 1.21 and 1.22 is called a number field or simply a field.

1.23. The order axioms. For every pair of elements x and y in R one (or both) of the relations x y (x is less than or equal to y) or y x has the following properties:

(a) x x for every x in R, and x y, y x together imply x = y;

(b) If x y, y z then x z.

(c) If x y, then x + z y + z for every z in R;

xyxy.

The relation x y can also be written in the form y x (y is greater than or equal to x). If x y and x ≠ y, we write x y (x is less than y) or y x (y is greater than x).

1.24. A set E R is said to be bounded from above if there exists an element z R such that x z for every x E, a fact expressed concisely by writing E z. Every number z with the above property relative to a set E is called an upper bound of E. An upper bound z0 of the set E is called the least upper bound of E if every other upper bound z of E is greater than or equal to z0 (why is z0 unique?). The least upper bound of E is denoted by sup E (from the Latin supremum). We can now state the following

Least upper bound axiom. Every set E R which is bounded from above has a least upper bound.

1.3. Consequences of the Addition Axioms

Our next task is to deduce various implications of the above axioms which will be needed later. We start with certain consequences of the addition axioms.

1.31. THEOREM. The system R contains a unique zero element.

Proof. Suppose R contains two zero elements 01 and 02. Then it follows from Axioms a and c of Sec. 1.21 that

†

1.32. THEOREM. Every element x in R has a unique negative.

Proof. Suppose x has two negatives y1 and y2, so that x + y1 = x + y2 = 0. Then it follows from Axioms a–c of Sec. 1.21 that

y2 = y2 +0 = y2 + (x +y1) =(y2 + x)+y1

= y1 + (x +y2) = y1 + 0 = y

The negative of the element x is denoted by −x. The sum x + (−y), written more concisely as x − y, is called the difference of x and y. The negative −(x + y) of the sum x + y is the sum of the negatives of x and y, since

x + y − x − y = x − x + y − y = 0 + 0 = 0.

1.33. THEOREM. The equation

has a unique solution in R, equal to b − a.

Proof. Adding the number −a to both sides of (1) and using Axioms a–c of Sec. 1.21, we find that

a + x − a = x + a − a = x + 0 = x = b − a,

so that the solution, if it exists, equals b − a. But b − a is a solution, since

a + (b − a) = a + b + (−a) = b + a + (−a) = b + 0 = b

1.4. Consequences of the Multiplication Axioms

1.41. a. THEOREM. The system R contains a unique unit element.

Proof. Suppose R contains two unit elements 11 and 12. Then it follows from Axiom a of Sec. 1.22 that

b. THEOREM. Every element x ≠ 0 in R has a unique reciprocal.

Proof. Suppose x has two reciprocals u1 and u2, so that xu1 = xu2 = 1. Then it follows from Axioms a–c of Sec. 1.22 that

1.42. The reciprocal of the element x is denoted by 1/x. The reciprocal 1/xy of the product xy is the product of the reciprocals of x and y, since

The product x · 1/z, written more concisely as x/z, is called the quotient (or ratio) of x and z.

1.43. Definition. The numbers

1, 2 = 1 + 1, 3 = 2 + 1, ..., n = (n − 1) + 1, ...

are called natural numbers (or positive integers). Thus the set of natural numbers can be defined as the smallest numerical set containing the number 1 and containing the number n + 1 whenever it contains the natural number n.

In many problems one must show that some numerical set A (e.g., the set of all natural numbers n for which some property Tn, depending on n, is valid) contains all natural numbers. The method of mathematical induction, used in such problems, consists in verifying that

(1) A contains 1;

(2) If A contains a natural number n, then A also contains n + 1.

If these two conditions are satisfied, it is clear from the foregoing that A contains all natural numbers. Thus the method of mathematical induction is a consequence of the very definition of the natural numbers.

1.44. a. By the integers we mean the natural numbers together with their negatives and the number zero.

b. Let m be an integer. Then the integer 2m is said to be even, while the integer 2m + 1 is said to be odd.

c. By the rational numbers we mean all quotients of the form m/n, where m and n are integers and n ≠ 0.

d. All other real numbers are said to be irrational.

1.45. THEOREM. The equation

has a unique solution in R, equal to b/a.

Proof. Dividing both sides of (1) by a and using Axioms a–c of Sec. 1.31, we find that

so that the solution, if it exists, equals b/a. But b/a is a solution, since

1.46. By definition,

and hence obviously

for arbitrary m, n = 1, 2, .... The expression xn is called the "nth power of x." To define xn for arbitrary integers, we set

for arbitrary x ≠ 0. We now verify that the formulas (2) continue to hold for arbitrary integers m and n. Suppose first that m 0, n = −q 0. Then

if q m, while

if q n, by Sec. 1.42. On the other hand, if m = −p 0, n = −q 0, then

again by Sec. 1.42. The second of the formulas (2) is proved similarly.

1.47. a. THEOREM. The formula

0 · x = 0

holds for every x in R.

Proof. Clearly

0 · x + 1 · x = (0 + 1)x = 1 · x = x,

0 · x + 1 · x = 0 · x + x,

and hence

x = 0 · x + x.

But then

0 · x = x − x = 0,

by †

It follows that 0 has no inverse, since the equation 0 · x = 1 is impossible. This justifies the high school rule: Don’t divide by zero.

b. On the other hand, if xy = 0 and x ≠ 0, then

Thus, if a product vanishes, so does at least one of its factors.

1.48. THEOREM. If u ≠ 0, ν ≠ 0, then

Proof. Merely note that

1.49. THEOREM. The formula

holds for every x in R.

Proof.† By Theorem 1.47a,

(−1) x + x = [(−1) + 1]x = 0 · x = 0,

which implies

The results proved in Secs. 1.3 and 1.4 guarantee that the real numbers satisfy all the familiar identities of elementary algebra (like the binomial theorem, the formulas for the sums of arithmetic and geometric progressions, and various formulas involving determinants).

1.5. Consequences of the Order Axioms

1.51. Order and addition

a. LEMMA. If x y, y z, and x = z, then x = y = z.

Proof. Since y z = x, we have y x and hence y = x by Axiom a of

b. LEMMA. ‡ If x y, y z, then x z. Similarly, if x y, y z, then x z.

Proof. An immediate consequence of

c. THEOREM. The following inequalities are equivalent:

0 yy − x, −y −x, x − y 0.

Proof. Adding −x to both sides of the first inequality and invoking Axiom c of Sec. 1.23, we get the second inequality. Similarly, adding −y to both sides of the second inequality, we get the third, adding x to both sides of the third inequality, we get the fourth, and, finally, adding y

d. LEMMA. If x y, then x + z y + z for every z in R.

Proof. Clearly x y implies x y and hence x + z y + z. But if x + z = y + z, then, adding −z to both sides, we get x = y which contradicts x y. It follows that x + z y + z

e. THEOREM. If xy1, ..., xn yn, then

x1 + ··· + xn y1 + ··· + yn,

where

x1 + ··· + xn y1 + ··· + yn,

if xj yj for at least one pair xj, yj.

Proof. By Axiom c of Sec. 1.23,

where, by the Lemma 1.51d, if xj yj for at least one pair xj, yj

Thus inequalities going in the same direction can be added. In particular, x0, ..., xn 0 implies s = x1 + ... + xn 0, where s 0, if xj 0 for at least one j.

f. THEOREM. The following inequalities are equivalent:

x yy – x,–y – x,x – y 0.

Proof. This follows from

1.52. Definition. A real number x is said to be nonnegative if x 0, positive if x 0, nonpositive if x 0, and negative if x 0.

Thus the number 0 is simultaneously nonnegative and nonpositive.

1.53. Definition. Given two real numbers x and y, suppose x y, say. Then x is called the minimum of the numbers x and y, and we write x = min{x, y}. By the same token, y is called the maximum of the numbers x and y, and we write y = max {x, y). Using induction, we can define min and max {x1, ..., xn} for any finite set of numbers x1, ..., xn, for example, by setting

max{x1, ..., xn = max {max {x1, ..., xn − 1}, xn}

The number

|x| = max {x, –x}

is called the absolute value or modulus of the number x. Thus |x| = x if x 0, while |x| = −x if x 0. The number |x| is nonnegative for every x, and |−x| = |x|.

1.54. a. THEOREM. If a 0, the inequality |xa is equivalent to the inequalities

Proof. Obviously x a if x 0, while −x a if x 0. Moreover, x = |xa if x 0, while −x |xa if x

Noting that −x a is equivalent to −a x (by Theorem 1.51c), we can write (1) in the form

b. THEOREM. The equation

holds for arbitrary real numbers x and y.

Proof. If x and y are both nonnegative or both nonpositive, the inequality follows at once from the definition of the absolute value. If, say, x 0, y 0, then

x + y x x + |y| = |x| + |y|,

−x − y −y = |y|x| + |y|,

so that

|x + y| = max {x + y, −x − y|x| + |y

c. Applying induction to (2), we get

|x1 + ... + xn|x1| + ... + |xn|.

1.55. Order and multiplication

a. LEMMA. If x 0, y 0, then xy 0.

Proof. Use Axiom d of

b. THEOREM. If x y, z 0, then xz yz.

Proof. We need only note that

yz − xz = (y − x) z 0,

by Axiom d of

c. It follows from in both places in Theorem 1.55b.

d. In particular, xx x 1, while xx if x 1.

e. If x yz u, then xz yz yu, so that inequalities can be multiplied under these conditions.

f. In particular, if x y, then xy², ..., xn yn.

g. THEOREM. If x 0, y 0, then xy 0, while if x 0, y 0, then xy 0.

Proof. In the first case −x 0. Hence, by Axiom d of Sec. 1.23 and Theorem 1.49,

(−x)y = (−1) xy = −(xy0,

which implies xy 0. In the second case −y 0, and hence −xy 0, xy

.

h. In particular, x² = x · x 0 for all x 0 and then from 2, etc.

i. THEOREM. The formula |xy| = |x||y| holds for all x and y.

Proof. If x 0, y 0, then |x| = x, |y| = y and xy 0, by Axiom d of Sec. 1.23, so that |xy| = xy = |x| |y|. Similarly, if x 0, y 0, then |x| = −x, |y| = −y and. xy 0, by Theorem 1.55g, so that |xy| = xy = (−x)(−y) = |x||y|. If x 0, y 0, then |x| = x |y| = −y and xy 0, by Theorem 1.55g again, so that |xy| = −xy = (−x) (−y) = |x||y|. If x 0, y 0, then |x| = x, |y| = −y and xy 0, by Theorem 1.55g again, so that |xy| = −xy = x (−y) = |x| |y| and similarly for the case x 0, y

1.56. THEOREM. If x 0, then 1/x 0. Moreover x y implies

Proof. The first assertion follows from

and x y by 1/xy

In particular, all rational numbers of the form p/q, where p and q are natural numbers, are positive.

1.57. The following principle is often used in proofs:

THEOREM. If a number z is nonnegative and less than every positive number, then z = 0.

Proof. If z 0, then, by hypothesis, z z

1.6. Consequences of the Least Upper Bound Axiom

1.61.† A set E R is said to be bounded from below if there exists an element z R such that z x for every x E, a fact expressed concisely by writing z E. Every number z with the above property relative to a set E is called a lower bound of E. If E is bounded from above, i.e., if there exists a number z such that E z, then −E (the set of all numbers −x with x E) is bounded from below, since x z implies −z −x. In particular, −z is a lower bound of the set −E. Conversely, if E is bounded from below by a number z, the same argument shows that −E is bounded from above by the number −z.

Suppose the set E is bounded from below. Then a lower bound z0 of E is called the greatest lower bound of E if every other lower bound z of E is less than or equal to z0 (why is z0 unique?). The greatest lower bound of E is denoted by inf E (from the Latin infimum).

THEOREM. Every set E R which is bounded from below has a greatest lower bound, equal to −sup (−E).

Proof. The set −E is bounded from above and hence has a least upper bound ξ = sup(−E), by the axiom of Sec. 1.24. If x E, then −x ξ and hence −ξ x, i.e., −ξ is a lower bound of the set E. Let η be any other lower bound of E. Then −η is an upper bound of −E, and hence, by the definition of the least upper bound, −η sup(−E) = ξ or equivalently η −ξ. In other words, inf E exists and equals −ξ = −sup(−E

1.62. a. THEOREM. If the sets E and F are bounded from above and E F, then supE supF, while if E and F are bounded from below and E F, then inf F inf E.

Proof. In the first case, sup F is an upper bound for F and hence all the more so for E F, so that sup E sup F. In the second case, inf F is a lower bound for F and hence all the more so for E F, so that inf F E

b. THEOREM. If x y for arbitrary x E, y F, then E is bounded from above, F is bounded from below, and sup E inf F.

Proof. The set E is bounded from above by any y F. Hence sup E exists, and sup E y for any y F. It follows that F is bounded from below by the number sup E, and hence that sup E inf F

1.63. Next we prove the existence and uniqueness of the "nth root" of any positive number.

THEOREM. Given any real x 0 and integer n 0, there exists a unique nth root of x, i.e., a number y 0 such that yn = x.

Proof (after W. Rudin). Let A be the set of all positive z such that zn x.

Then A is bounded from above (by 1 if x 1 and by x if x 1). Let

We now show that yn = x, thereby proving the theorem.

Suppose yn x and let x − yn = ε. Then, by the binomial theorem,

for any positive h 1. Choosing

we get (y + h)n yn + ε = x, which contradicts the definition (1). Therefore yn x. Suppose yn x and let yn − x = ε. Then

for any positive h 1. Again choosing

we get (y − h)n yn − ε = x, which again contradicts (1). It follows that yn = x, as asserted. The uniqueness of the nimplied by yy

The nth root of x .

1.64. THEOREM. The formula

holds for arbitrary positive x and y.

Proof. Since ξn = x, ηn = y, we have (ξη)n = ξnηn

= xy = τn. But then

by the uniqueness of the n

Similarly, it can be shown that

for arbitray x 0 and integers m, n 1.

1.65. Suppose n is an even integer. Then (−x)n = (−1)nxn = xn 0 for all x ≠ 0, so that the equation yn = x 0 has both a positive solution yand a negative solution y, while the equation yn = x 0 has no real solutions at all. On the other hand, if n is an odd integer, the equation yn = x 0 has a unique real solution y and the equation yn = x 0 also has a unique solution, namely y .

Formulas (2) and (3) guarantee the validity for the real numbers of all the usual elementary algebraic results involving radicals (like the formulas for the solution of quadratic and cubic equations).

1.66. A set E R is said to be bounded from both sides, or simply bounded, if it is bounded both from above and from below. Every bounded set E has both a least upper bound sup E and a greatest lower bound inf E. The following are particularly important examples of bounded sets:

1.67. Intervals. The set of all real numbers x satisfying the inequality a x b is denoted by [a, b] and called a closed interval, with left-hand endpoint a and right-hand end point b (it is assumed that a b). The set of all real numbers x satisfying the inequality a x b is denoted by (a, b) and called an open interval, again with left-hand end point a and right-hand end point b. Thus the end points of a closed interval belong to the interval itself, while the end points of an open interval do not. In any event,

sup[a, b] = sup(a, b) = b,inf[a, b] = inf(a, b) = a.

It is also convenient to introduce half-closed and half-open intervals. Thus the sets {x: a x b} = (a, b] and {x: a x b} = [a, b) are both called intervals, the former half-open on the left and half-closed on the right, the latter half-closed on the left and half-open on the right. For completeness, we will sometimes regard a single point a as a closed interval, writing

{a} = [a, a] = {x: a x a}.

1.7. The Principle of Archimedes and Its Consequences

1.71. THEOREM (Principle of Archimedes) † Given arbitrary real numbers x 0 and y, there exists an integer n such that (n − 1) x y nx.

Proof. Suppose px y for every integer p. Then the set A of all numbers px is bounded from above, with y as an upper bound. It follows from the axiom of Sec. 1.24 that A has a least upper bound ξ = sup A. Since the number ξ − x ξ is no longer an upper bound of A, there exists an integer p such that px ξ − x. But then (p + 1)x ξ, so that ξ cannot be an upper bound of A. This contradiction proves the existence of an integer p such that px y. An analogous argument involving lower bounds shows that there exists an integer q such that qx y, where clearly q p. Examining all the pairs (q, q + 1), (q + 1, q + 2), ..., (p − 1, p), we find one among them, say (n − 1, n), such that (n − 1) x y while y nx

In particular, choosing x = 1, we find that there exists an integer n such that n y n for any given y R. The number n − 1 is called the integral part of y and is denoted by [y], while the number y − [y] is called the fractional part of y and is denoted by (y). Thus every number y is the sum

y = [y] + (y)

of its integral and fractional parts.

1.72. THEOREM. Given arbitrary real numbers x 1, y 0, there exists an integer n such that xn y xn.

Proof. This is the multiplicative version of the principle of Archimedes, and is proved by going over from integral multiples‡ of x to integral powers of x

1.73. THEOREM. Given arbitrary real numbers x 0 and y 0, there exists an integer n 0 such that

Proof. As in Theorem 1.71, y nx, where now y and

are positive. Multiplying both sides of (2) by x/n

In particular, it follows that

for any y 0. In fact, the set in curly brackets consists of positive numbers only and hence has a nonnegative greatest lower bound. But, as just shown, this greatest lower bound cannot be positive, and hence must equal zero.

1.74. COROLLARY. Each of the following systems of half-open intervals has an empty intersection:

Proof. If the intervals of the system (4) had a common point ξ, then ξ − a would be a common point of the system (3), while if the intervals of the system (5) had a common point η, then a − η

The corollary clearly remains true if y/n (n = 1, 2, ...) is replaced by y/nx(x 0 arbitrary) or by y/xn − ¹ (x 1 arbitrary), in particular, by y/10n − ¹.

1.75. THEOREM. Every open interval (a, b) contains a rational point.

Proof. Let h = b − a 0, and let n be an integer greater than 1/h (the existence of n follows from Theorem 1.71), so that 1/n h. By Theorem 1.71 again, there exists an integer m such that

where clearly

and hence

It follows that

i.e., the rational number

belongs to the interval (a, b

There are actually an infinite number of rational points in (a, b). In fact, applying the above theorem to the interval

gives a new rational number p/q such that

and this process can clearly be continued indefinitely.

1.76. THEOREM. Given any real number ξ, let Nξ be the set of all rational numbers s ξ, and let Pξ be the set of all rational numbers r ξ. Then

sup Nξ = ξ = inf Pξ. †

Proof. Let α = sup Nξ. Then, since s ξ for every s Nξ, we have α ξ by the very definition of the least upper bound. Suppose α ξ. By Theorem 1.75, there is a rational point p in (α, ξ). Since p ξ, we have p Nξ, which implies p sup Nξ = α, contrary to the condition p (α, ξ). Therefore the inequality α ξ is impossible, and hence α = sup Nξ = ξ. The fact that inf Pξ = ξ

1.77. Decimal representation of real numbers. Next we show how an arbitrary real number ξ can be represented by a suitable sequence of the symbols (digits) 0, 1, 2, ..., 9.

a. Suppose ξ 0. By Theorem 1.72, there exists a (unique) integer p such that

10p ξ 10p + ¹.

Having found the exponent p, we next find a number θ0 (from the set 1, 2, ..., 9) such that

θ0 · 10p (θ0 + 1) · 10p

(where 9 + 1 = 10). The number θ0 is also uniquely determined, since the intervals.

θ · 10p x (θ + 1) · 10p(θ = 0, 1, ..., 9)

are nonintersecting. Next, having found θ0, we find a number θ1 (from the set 0, 1, ..., 9) such that

θ0 · 10p + θ1 · 10p θ0 · 10p + (θ1 + 1) · 10p − ¹.

Continuing this process indefinitely, we get a sequence of symbols (digits from 0 to 9)

The underlying presence of the number p is indicated as follows: If p 0, we put a decimal point between the symbols θp and θp + 1 while if p 0, i.e., if p = −q, q 0, we write q additional zeros in front of the sequence (6) and then put a decimal point after the first zero. In this way, the influence of the number p is reflected in the expression (6).

Thus to every real number ξ 0 there corresponds, in accordance with this rule, an expression of the form (6) (possibly preceded by a string of zeros), with a decimal point in some position. The expression (6) is called the decimal representation or expansion of ξ, with (decimal) digits θ0, θ1, θ2, ... (in that order). The decimal representation of the number 1 is just 1.000 ..., and similarly for the numbers 2, 3, ..., 9. Similarly, the decimal representation of the number 10 is 10.000 ..., while for numbers of the form

(rational decimals), and only for such numbers, there are no more than t nonzero digits after the decimal point.

b. THEOREM. It is impossible for all the digits in an expression of the form (6) to be nines starting from some position, i.e., (6) cannot have an "infinite run of nines."

Proof. The presence of an infinite run of nines beginning with some number n after the decimal point (the "nth decimal place") would mean that the number ξ lies in all the intervals

But this is impossible, since the system of intervals

c. THEOREM. Let

be an arbitrary sequence of digits from 0 to 9, with a decimal point in some position, where not all the τi are zero and there are digits other than 9 arbitrarily far from the decimal point. Then there exists areal number ξ 0 with (8) as its decimal expansion.

Proof. Let τm be the first nonzero digit in (8). The decimal point is either to the right of τm by q 0 digits (not including τm), or to the left by t 1 digits (including τm); in the second case we set q = −t. We now show that the decimal expansion of the number ξ, defined as

coincides with (8). Let s be a fixed positive integer, choose r s such that τm + r 8, and let k r. Then, summing a geometric progression, we get

It follows that

and hence

10q · τm + ··· + 10q − s · τm + s 10q · τm + ··· +