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