suprem

Suprem, usually written supremum and often abbreviated as sup, is a fundamental concept in order theory and real analysis describing the least upper bound of a set. Let A be a nonempty subset of an ordered set (P, ≤). An element s in P is the supremum of A if (1) s is an upper bound of A (every a in A satisfies a ≤ s), and (2) s is the least such upper bound: for any u in P with a ≤ u for all a in A, we have s ≤ u. If such an s exists, it is unique.

Existence of a supremum depends on the ambient order. In the real numbers with the usual order, every nonempty subset that is bounded above has a supremum. If A has a maximum element, then that maximum equals the supremum. The supremum may or may not belong to A itself; for example, the open interval A = (0, 1) has supremum 1, but 1 is not an element of A. By contrast, the finite set A = {2, 3} has supremum 3, which is also its maximum.

Supremum is dual to infimum, the greatest lower bound, and is often denoted infimum as inf. The concept extends beyond the real numbers to general partially ordered sets, where not all subsets have a supremum. In complete lattices and in the real numbers, many important limits and convergence arguments use supremum to formalize the idea of “least upper bound.”