Page 163 - Vector Analysis

§A.1 Properties of Real Numbers 159

Remark A.2. The least upper bound and the greatest lower bound of S need not be a
member of S.

Remark A.3. The reason for defining sup H = ´8 and inf H = 8 is as follows: if
H ‰ A Ď B, then sup A ď sup B and inf A ě inf B.

( ( A ) B)

inf B inf A sup A sup B

Since H is a subset of any other sets, we shall have sup H is smaller then any real number,
and inf H is greater than any real number. However, this “definition” would destroy the
property that inf A ď sup A.

The “definition” of sup H and inf H is purely artificial. One can also define sup H = 8
and inf H = ´8.

Definition A.4. An open interval in R is of the form (a, b) which consists of all x P R Q
a ă x ă b. A closed interval in R is of the form [a, b] which consists of all x P R Q a ď
x ď b.

Proposition A.5. Let S Ď R be non-empty. Then

1. b = sup S P R if and only if

(a) b is an upper bound of S.
(b) @ ε ą 0, D x P S Q x ą b ´ ε.

2. a = inf S P R if and only if

(a) a is a lower bound of S.
(b) @ ε ą 0, D x P S Q x ă a + ε.

Proof. “ñ” (a) is part of the definition of being a least upper bound.

(b) If M is an upper bound of S, then we must have M ě b; thus b ´ ε is not an upper
bound of S. Therefore, D x P S Q x ą b ´ ε.

“ð” We only need to show that if M is an upper bound of S, then M ě b. Assume the

contrary. Then D M such that M is an upper bound of S but M ă b. Let ε = b ´ M ,

then there is no x P S Q x ą b ´ ε. ÑÐ ˝