
 Definition of Upper Bound and Least Upper Bound (Supremum) (Site not responding. Last check: 20071021) 
  The least upper bound, called the supremum, of a set S, is defined as a quantity M such that no member of the set exceeds M, but if ε is any positive quantity, however small, there is a member that exceeds M  ε. 
  The least upper bound of a function, f, is defined as a quantity M such that f(x) ≤ M for all x in its domain, but if ε is any positive quantity, however small, there is an x in the domain such that f(x) exceeds M  ε. 
  Let S be the supremum of A. By the definition of supremum, given any small positive number, ε, you can find an element of the set (and thus an element S 
 mcraefamily.com /MathHelp/CalculusLimitUpperBound.htm (457 words) 
