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cbcarey
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bounded above
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 sepsilon is less than a

