function). Hint: Use the triangle inequality to show that, for any pair of vectors

x and y, " #x# 9 #y# " - #x 9 y#.

6. BASIC PROPERTY OF REAL NUMBERS

A set S in RL is said to be bounded if there exists a positive number M such

that #x# - M for all x + S. Another way of stating this is S : B(0, M). The

number M is said to be a bound for the set. Note that if a set is bounded, then

there are in¬nitely many bounds.

We now shall deal with sets in R only. In R a set S being bounded means

that, for every x in S, "x" - M, or 9M - x - M. A set S in R is said to be

bounded above if there exists a real number A such that x - A for all x in S. A

set S is bounded below if there exists a real number B such that x . B for all x

in S. The number A is said to be an upper bound of S, the number B a lower

bound.

12 TOPICS IN REAL ANALYSIS

A number U is said to be a least upper bound (l.u.b.), or supremum (sup), of

a set S

(i) if U is an upper bound of S and

(ii) if U is another upper bound of S and then U . U.

If a set has a least upper bound, then the least upper bound is unique. To

see this, let U and U be two least upper bounds of a set S. Then since U is

an upper bound and U is a least upper bound, U . U . Similarly, U . U ,

and so U : U .

Condition (ii) states that no number A : U can be an upper bound of S.

Therefore condition (ii) can be replaced by the equivalent statement:

(ii) For every 9 0 there is a number x in S with U . x 9 U 9 .

C C

Let S be the set of numbers 1 9 1/n, n : 1, 2, 3, . . . . Then one is the l.u.b. of

this set. Note that one does not belong to S. Now consider the set S : S 6 +2,.

The number 2 is the l.u.b. of this set and belongs to S . The reader should be

clear about the distinction between an upper bound and a supremum, or l.u.b.

A number L is said to be a greatest lower bound (g.l.b.), or in¬mum (inf), of

a set S

(i) if L is a lower bound of S and

(ii) if L is another lower bound of S and then L - L .

Observations analogous to those made following the de¬nition of l.u.b. hold

for the de¬nition of g.l.b. We leave their formulation to the reader.

Exercise 6.1. Let L : g.l.b. of a set S. Show that there exists a sequence of

points +x , in S such that x ; L . Show that the sequence +x , can be taken to

I I I

be nonincreasing, that is, x - x for every k. Does L have to be a limit point

I> I

of S?

Exercise 6.2. Let S be a set in R. We de¬ne 9S to be +x : 9x + S,. Thus 9S

is the set that we obtain by replacing each element x in S by the element 9x.

Show that S is bounded below if and only if 9S is bounded above. Show that

is the g.l.b. of S if and only if 9 is the l.u.b. of 9S.

We can now state the basic property of the real numbers R, which is

sometimes called the completeness property.

BASIC PROPERTY OF REAL NUMBERS 13

Basic Property of the Reals

above l.u.b.

Every set S : R that is bounded has a .

below g.l.b.

It follows from Exercise 6.2 that every set in R that is bounded above has

a l.u.b. if and only if every set that is bounded below has a g.l.b. Thus, our

statement of the completeness property can be interpreted as two equivalent

statements.

Not every number system has this property. Consider the rational numbers

Q. They possess all of the algebraic and order properties of the reals but

do not possess the completeness property. We will outline the argument

showing that the rationals are not complete. For details see Rudin [1976].

Recall that a rational number is a number that can be expressed as a quotient

of integers and recall that (2 is not rational. (The ancient Greeks knew that

(2 is not rational.) Consider the set S of rational numbers x de¬ned by S :

+x : x rational, x : 2,. Clearly, 2 is an upper bound. It can be shown that for

any x in S there is an x in S such that x 9 x. Thus no element of S can be an

upper bound for S, let alone a l.u.b. It can also be shown that for any rational

number y not in S that is an upper bound of S there is another rational number

y that is not in S, that is, also an upper bound of S and satis¬es y : y. Thus

no rational number not in S can be a l.u.b. of S. Hence, there is no l.u.b. of S

in the rationals. The number (2 in the reals is the candidate for the title of

l.u.b., but (2 is not in the system of rational numbers.

Exercise 6.3. Let A and B be two bounded sets of real numbers with A 3 B.

Show that

sup+a : a + A, - sup+b : b + B,,

inf+a : a + A, . inf+b : b + B,.

Exercise 6.4. Let A and B be two sets of real numbers.

(i) Show that if A and B are bounded above, then

sup+(a ; b) : a + A, b + B, : sup+a : a + A, ; sup+b : b + B,.

(ii) Show that if A and B are bounded below, then

inf+(a ; b) : a + A, b + B, : inf+a : a + A, ; inf+b : b + B,.

Exercise 6.5. Let a and b be real numbers and let I : +x : a - x - b,. A real-

valued function f is said to have a right-hand limit at a point x , where

a - x : b, if for every 9 0 there exists a ( ) such that whenever 0 : x9x

14 TOPICS IN REAL ANALYSIS

: ( ) the inequality " f (x) 9 " : holds. We shall write

f (x ;) : lim f (x) : .

VV>

(i) Formulate a de¬nition for a left-hand limit at a point x , where

a : x - b. If is the left-hand limit, we shall write

f (x 9) : lim f (x) : .

VV\

(ii) Show that f has a limit at a point x , where a : x : b, if and only if

f has right- and left-hand limits at x and these limits are equal.

Exercise 6.6. Let I be as in Exercise 6.5. A real-valued function f de¬ned on I

is said to be nondecreasing if for any pair of points x and x in I with x 9 x

we have f (x ) . f (x ).

(i) Formulate the de¬nition of nonincreasing function.

(ii) Show that if f is a nondecreasing function on I, then f has a right-hand

limit at every point x, where a - x : b, and left hand limit at every

point a : x - b.

(iii) Show that if I is an open interval (a, b) : +x : a : x : b, and if f is

nondecreasing and bounded below on I, then lim f (x) exists. If f

V?>

f (x) exists.

is bounded above, then lim

V@\

7. COMPACTNESS

A property that is of great importance in many contexts is that of compactness.

The de¬nition that we shall give is one that is applicable in very general

contexts as well as in RL. We do this so that the reader who takes more

advanced courses in analyis will not have to unlearn anything. After presenting

the de¬nition of compactness, we will state, without proof, two theorems that

give necessary and suf¬cient conditions for a set S in RL to be compact. We

frequently shall use the properties stated in these theorems when working with

compact sets.

Let S be a set in RL. A collection of open sets +O , is said to be an open

? ?Z

cover of S if every x + S is contained in some O . A set S in RL is said to be

?

compact if for every open cover +O , of S there is a ¬nite collection of sets

? ?Z

O , . . . , O from the original collection +O , such that the ¬nite collection

? ?Z

K

O , . . . , O is also an open cover of S. In mathematical jargon this property is

K

stated as ˜˜Every open cover has a ¬nite subcover.™™

To illustrate, consider the set S in R de¬ned by S : (0, 1) : +x : 0 : x : 1,.

For each positive integer k, let O : (0, 1 9 1/k) : +x : 0 : x : 1 9 1/k,. Then

I

+O , is an open cover of S, but no ¬nite collection of sets in the cover will

I I

COMPACTNESS 15

cover S. Thus, (0, 1) is not compact. It is dif¬cult to give a meaningful example

of a compact set based on the de¬nition, since we must show that every open

cover has a ¬nite subcover.

T 7.1. In RL a set S is compact if and only if every sequence +x , of points

I

in S has a subsequence +x , that converges to a point in S (Bolzano ”Weierstrass

IH

property).

T 7.2. In RL a set S is compact if and only if S is closed and bounded.

We refer the reader to Bartle and Sherbert [1999] or Rudin [1976] for

proofs of these theorems.

We emphasize that the criteria for compactness given in Theorems 7.1 and

7.2 are not necessarily valid in contexts more general than RL.

Theorem 7.2 provides an easily applied criterion for compactness. Using

Theorem 7.2, we see immediately that S : (0, 1) : +x : 0 : x : 1, is not com-

pact in R. (S is not closed.) We also see that S : [0, 1] : +x : 0 - x - 1, is

compact, since it is closed and bounded.

We conclude with another characterization of compactness that is valid in

more general contexts than RL. A set S is said to have the ¬nite-intersection

property if for every collection of closed sets +F , such that 7F 3 cS there

? ?

is a ¬nite subcollection F , . . . , F such that 7I F 3 cS. We shall illustrate

G G

I

the de¬nition by exhibiting a set that fails to have the ¬nite-intersection

property. The set (0, 1) : +x : 0 : x : 1, in R does not have the ¬nite-

intersection property. Take F : [191/n, -), n : 2, 3, 4, . . . . Then 7 F :

L L

L

[1, -) : c(0, 1), yet the intersection of any ¬nite subcollection of the F ™s has

L

nonempty intersection with (0, 1) and so is not contained in c(0, 1).

T 7.3. A set S is compact if and only if it has the ¬nite-intersection

property.

The proof is an exercise in the use of Lemma 3.1. Let S have the ¬nite-

intersection property. Let +O , be an open cover of S. Then S 3 8 O :

? ?

c7 (cO ), and so cS 4 c(c7 (cO )) : 7 (cO ). Each set cO is closed, so

? ? ? ?

there exist a ¬nite number of sets O , . . . , O such that cS 4 7I (cO ). Hence

G

I G

S 3 c (7 I (cO )) : 8L O , and so +O , has a ¬nite subcover O , . . . , O .

G G G

G ? I

Thus we have shown that S is compact. Now let S be compact. Let +F , be a

?

collection of closed sets such that 7 F 3 cS. Then c(8 (cF )) 3 cS, and so

? ?

8 (cF ) 4 S. Thus +(cF ), is an open cover of S. Since S is compact, there exist

? ?

a ¬nite subcollection of sets F , . . . , F such that 8I (cF ) 4 S. Therefore

G

I G

7I F : c 8I (cF ) 3 cS, and thus S has the ¬nite-intersection property.

G G G G

Remark 7.1. Note that the ¬nite-intersection property can be stated in the

following equivalent form. Every collection of closed sets +F , such that

?

(7 F ) 5 S:

has a ¬nite subcollection F , . . . , F such that (7I F ) 5S:

.

G G

?? I

16 TOPICS IN REAL ANALYSIS

8. EQUIVALENT NORMS AND CARTESIAN PRODUCTS

A norm on a vector space D is a function (·) that assigns a real number (v)

to every vector v in the space and has the properties (1)”(3) and (5) listed in

Section 2, with #·# replaced by (·). Once a norm has been de¬ned, one can

de¬ne a function by the formula (v, w) : (v 9 w). Properties (1)”(3) and

(5) of the norm imply that the function satis¬es the properties (1)”(3) of a

metric and therefore is a metric, or distance function. Once a distance has been

de¬ned, open sets are de¬ned in terms of the distance, as before. Convergence

is de¬ned in terms of open sets.

To illustrate these ideas, we give an example of another norm in RL, de¬ned