A number b is called an upper bound for a set S of numbers if x s b for all x in S. For example 5, 6.5, and 13 are upper bounds for the set S = {1,2,3, 4, 5}. The number 5 is the least upper bound for S (the smallest of all upper bounds). Similarly, 1.6, 2, and 2.5 are upper bounds for the infinite set T = {1.4, 1.49, 1.499. 1.4999.... }, whereas 1.5 is its least upper bound. Find the least upper bound of each of the following sets.
(a) S = {- 10, -8, -6, -4, -2}
(b) S = {- 2, - 2.1, -2.11, -2.111, - 2.1111,....}
(c) S = {2.4, 2.44, 2.444, 2.4444,...}
(d) S = {1 - ½, 1 - 1/3, 1 - ¼, 1 - 1/5,...}
(e) S = {x:x = (-1)n + 1/n, n a positive integer}; that is, S is the set of all numbers x that have the form x = (- 1)n + 1/n, where n is a positive integer.
(f) S = {x:x2 < 2, x a rational number}