Least Upper Bound

Let  be a nonempty set of real numbers that has an upper bound. Then a number  is called the least upper bound (or the supremum, denoted ) for  iff it satisfies the following properties:

1.  for all .

2. For all real numbers , if  is an upper bound for , then .

REFERENCES:

Rudin, W. Principles of Mathematical Analysis, 3rd ed. New York: McGraw-Hill, p. 4, 1976.