Least-upper-bound propertyIn mathematics, the least-upper-bound property (sometimes called completeness or supremum property or l.u.b. property) is a fundamental property of the real numbers. More generally, a partially ordered set X has the least-upper-bound property if every non-empty subset of X with an upper bound has a least upper bound (supremum) in X. Not every (partially) ordered set has the least upper bound property. For example, the set of all rational numbers with its natural order does not have the least upper bound property.
Provence wineProvence (Provençal) wine comes from the French wine-producing region of Provence in southeast France. The Romans called the area provincia nostra ("our province"), giving the region its name. Just south of the Alps, it was the first Roman province outside Italy. Wine has been made in this region for at least 2,600 years, ever since the ancient Greeks founded the city of Marseille in 600 BC. Throughout the region's history, viticulture and winemaking have been influenced by the cultures that have been present in Provence, which include the Ancient Greeks, Romans, Gauls, Catalans and Savoyards.
Infimum and supremumIn mathematics, the infimum (abbreviated inf; plural infima) of a subset of a partially ordered set is the greatest element in that is less than or equal to each element of if such an element exists. In other words, it is the greatest element of that is lower or equal to the lowest element of . Consequently, the term greatest lower bound (abbreviated as ) is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset of a partially ordered set is the least element in that is greater than or equal to each element of if such an element exists.
