Completeness

• Why Does a Nonempty Bounded Set of Reals Have a Supremum?
  A nonempty set of real numbers that is bounded above has a supremum because the real numbers satisfy the least-upper-bound property.
• Does Every Nonempty Bounded-Above Set of Real Numbers Have a Supremum?
  Every nonempty subset of the real numbers that is bounded above has a supremum, by the completeness property of the real numbers.