REAL NUMBERS, R

10. BOUNDED AND UNBOUNDED SETS

Let A be a set of numbers, then A is called bounded set if A is the subset of a finite interval. An equivalent definition of boundedness is;

Definition 3.1: Set A is bounded if there exists a positive number M such that

|x | < = M.

for all x £ A. A set is called unbounded if it is not bounded

Notice then, that A is a subset of the finite interval [-M, M].

Example 4.1: Let A = {1, ½, 1/3…}. Then A is bounded since A is certainly a subset of the closed interval [0, 1].

Example 4.2: Let A = {2, 4, 6,…..}. Then A is an unbounded set.

Example 4.3: Let A = {7, 350, -473, 2322, 42}. Then A is bounded

Remark 3.3: If a set A is finite then, it is necessarily bounded.

If a set is infinite then it can be either bounded as in example 4.1 or unbounded as in example 4.2