Suppose that A <;;; B are nonempty subsets of R.

(i) If B has a supremum, then sup A ~ sup B.

(ii) If B has an infimum, then inf A 2:: inf B.

PROOF. (i) Since A <;;; B, any upper bound of B is an upper bound of A.

Therefore, sup B is an upper bound of A. It follows from the Completeness Axiom that sup A exists, and from Definition 1.16iii that sup A ~ sup B.