Page 162 - Vector Analysis

Appendix A
Appendix

A.1 Properties of Real Numbers

Definition A.1. Let H ‰ S Ď R. A number M P R is called an upper bound (上界) for
S if x ď M for all x P S, and a number m P R is called a lower bound (下界) for S if
x ě m for all x P S. If there is an upper bound for S, then S is said to be bounded from
above, while if there is a lower bound for S, then S is said to be bounded from below.
A number b P R is called a least upper bound (最小上界) if

1. b is an upper bound for S, and

2. if M is an upper bound for S, then M ě b.
A number a is called a greatest lower bound (最大下界) if

1. a is a lower bound for S, and

2. if m is a lower bound for S, then m ď a.

•( S )•
m M

an lower bound for S an upper bound for S

If S is not bounded above, the least upper bound of S is set to be 8, while if S is not
bounded below, the greatest lower bound of S is set to be ´8. The least upper bound of
S is also called the supremum of S and is usually denoted by lubS or sup S, and “the”
greatest lower bound of S is also called the infimum of S, and is usually denoted by glbS
or inf S. If S = H, then sup S = ´8, inf S = 8.

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