Least upper bound

• How to Prove That sup(A ∪ B) = max{sup A, sup B}
  If A and B are nonempty and bounded above, the supremum of their union is the larger of their individual suprema. Here is the least-upper-bound proof.
• How to Calculate the Supremum of a Set
  To calculate a supremum, find an upper bound and prove set elements come arbitrarily close to it; unlike a maximum, the supremum need not belong to the set.