higher analysis

a set A in R is bounded above if there exists a number b in R such that a is less than or equal to b for all a in A

• A real number s is the least upper bound for a set A in R if it meets the following two criteria:
• (i) s is an upper bound for A
• (ii) if b is any upper bound for A, then s is less than or equal to b

every nonempty set of real numbers that is bounded above has a least upper bound

assume s is in R is an upper bound for a set A in R. Then, s=supA if and only if, for every choice of epsilon greater than 0, there exists an element a in A satisfying s-epsilon is less than a