Convergent sequence

A sequence in a topological space converges to a point when, for every neighborhood of that point, all terms beyond some index lie in that neighborhood.

Neighborhood (topology): A set containing an open set around a point in a topological space. Convergence requires every neighborhood of the proposed limit to contain all sufficiently late terms.
Topological space: A set equipped with a collection of open subsets satisfying the axioms of a topology. Neighborhood-based convergence is defined using the topology of the space.
Uniqueness of limits: The theorem that a convergent sequence has at most one limit in a Hausdorff space. Distinct limits are impossible when disjoint neighborhoods can separate their points.
Divergent sequence: A sequence that does not converge to any point of its space. It fails the requirement that every neighborhood of some point eventually contain all terms.
Limit of a sequence: A point that a sequence approaches in the sense specified by its space's structure. The limit is the point whose every neighborhood eventually contains the sequence.
Open set: A subset of a topological space that is a member of its topology. Open sets provide the basic pieces from which neighborhoods are formed.
Continuous function: A function for which inverse images of open sets are open. Continuous functions send convergent sequences to sequences converging to the image of the limit.
Bounded sequence: A sequence whose terms lie within a bounded region of a metric space or normed space. Boundedness alone does not ensure that the terms approach a limit.
Metric space: A set equipped with a distance function satisfying nonnegativity, identity, symmetry, and the triangle inequality. In metric spaces, distance-based convergence is equivalent to the neighborhood definition.
Natural numbers: The counting numbers, commonly used to index sequences. The phrase “eventually” means that a property holds at every index after some natural-number index.

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