limitants

Limitants is a term found in some mathematical texts to denote the set of limit values that a sequence, function, or more general indexing process can approach within a given space. It is closely related to the concepts of limit points or cluster points, but is used in a more specific or alternative notation in certain sources.

Definition and scope. For a sequence x_n in a topological space X, a point x is called a limitant of the sequence if there exists a subsequence x_{n_k} that converges to x. The collection of all such points forms the limitant set of the sequence. When nets or filters are involved, the corresponding limitants are the limits of convergent subnets or subfilters. Thus, limitants generalize the idea of what values can be approached by the process, not just the single value to which the full sequence may converge.

Basic properties. The limitant set is typically closed. In many common spaces (for example, compact or complete spaces) the limitant set is nonempty for bounded or relatively compact families. If a sequence converges, its limitant set reduces to a singleton containing the usual limit. In Euclidean spaces, the limitant set of a bounded sequence is compact.

Examples. The sequence x_n = (-1)^n in R has limitant set {-1, 1}. The constant sequence x_n = 2 has limitant {2}.

Relation to standard terminology. A limit is a special case of a limitant when a sequence or net converges to a single value. The term limitant is not universally standardized and is often superseded by standard terms such as limit points or cluster points in many texts.