
 Definition of Upper Bound and Least Upper Bound (Supremum) 
  S. The set S is said to be "bounded above" by C. A function, f, is said to have a upper bound C if f(x) ≤ C for all x in its domain. 
  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. 
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