Definitions: Let A be a non-empty subset of the real numbers (R). A is bounded above if there exists a real number ̱ such that a ≤ ̱ for all a ∈ A. In this case, the value ̱ is called an upper bound of A. A is bounded below if there exists a real number ̱ such that a ≥ ̱ for all a ∈ A. In this case, ̱ is called a lower bound for A. The value α is the least upper bound (lub) or supremum (sup) for a set A if (1) α is an upper bound for A and (2) if λ < α, then λ is not an upper bound for A. (2) says if η is any upper bound for A, then α ≤ η. Similarly, a value β is the greatest lower bound (glb) or infimum (inf) of A if (1) β is a lower bound for A, and (2) if λ > β, then λ is not a lower bound for A. (2) says that if η is any lower bound for A, then β ≥ η.
The least upper bound and greatest lower bound are unique if they exist. We are assuming that the real numbers has the least upper bound property: If A is any non-empty subset of the reals that is bounded above, then A has a least upper bound. We can show that the real numbers also has the greatest lower bound property, as well.
The problem:
Let A and B be non-empty subsets of the real numbers such that for all a in A and for all b in B, a < b. Show that lub(A) and glb(B) exist, and that lub(A) ≤ glb(B).