Borné

Borné (masculine) or bornée (feminine) is a French adjective meaning bounded, confined, or limited. In mathematics and related disciplines, it describes objects whose values lie within a finite extent. A set S in a metric or normed space is called bounded if there exists a point x0 and a real number R such that the distance d(x, x0) is at most R for every x in S. Equivalently, S has finite diameter. A function f defined on a domain D is bounded if there exists a real number M with |f(x)| ≤ M for all x in D. A sequence (x_n) is bounded if its terms stay within some fixed interval. By contrast, a sequence or set that is not bounded is called unbounded. The unit interval [−1, 1] is bounded; the sequence x_n = n is unbounded.

Etymology: borné derives from the verb borner, to bound or enclose; the noun borne denotes a boundary. In colloquial French, the adjective can describe a person who is narrow-minded or limited in outlook, as in c'est borné, a pejorative remark.

See also: boundedness, unboundedness, and related notions in analysis such as compactness and completeness. In mathematical practice, distinguishing between bounded and unbounded objects helps characterize convergence, continuity, and the behavior of functions and sequences.