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limit point[′lim·ət ‚pȯint]
(or accumulation point). A limit point of a set A in a metric space is a point ξ in a space such that arbitrarily close to ξ there are points in A distinct from ξ. In other words, ξ is a limit point if any neighborhood of ξ contains an infinite number of points in A. A characteristic property of ξ is the existence of at least one sequence of distinct points of A that converges to ξ.
A limit point of a set does not have to belong to the set. Thus, every point on the real axis is a limit point for the set of rational points, because for every number—rational or irrational—we can find a sequence of distinct rational numbers that converges to it. Not every infinite set has a limit point; the set of integers, for example, lacks such a point. Every infinite bounded set of a Euclidean space, however, has at least one limit point.