Convergence and closed sets

The error tolerance is chosen first, then a cutoff index may be selected in response. Beyond that cutoff every term must satisfy the same tolerance. The cutoff can depend on the tolerance, but it cannot be chosen separately for each late term. For x_n=1/n in the real line, choosing an integer N larger than 1/epsilon gives the required bound for every n at least N. Alternating signs do not prevent convergence when the magnitude tends to zero; the distance, not the direction of approach, decides the question.

A sequence may enter a small ball infinitely often without converging to its center. Alternating between zero and one repeatedly visits every neighborhood of zero but also repeatedly leaves neighborhoods of radius less than one. Such a sequence has a subsequence converging to zero, which is a weaker claim. Convergence also implies boundedness: a late tail lies in a unit ball around its limit, and the finitely many earlier distances have a finite maximum. This implication is useful, but boundedness alone cannot force convergence.

Suppose the same sequence converges to two points x and y. The triangle inequality bounds their fixed distance by the sum of the two distances from a sufficiently late term. Each summand can be made arbitrarily small, so the fixed distance must be zero. The metric's identity-of-indiscernibles condition then gives x=y. This argument explains why that condition matters: in a pseudometric, distinct zero-distance points can both serve as limits of the same sequence.

A set is closed when its complement is open. In a metric space this is equivalent to containing every ambient limit of its own convergent sequences. For one direction, an excluded limit would have a ball in the open complement, contradicting eventual membership in that ball. For the reverse direction, if the complement is not open at an outside point, choose a set point within 1/n of that point for every n. The chosen sequence converges outside and violates the proposed sequential condition.

The interval (0,1) is not closed in the real line because 1/n, for n at least two, converges to the missing point zero. Yet the whole space X=(0,1) is closed in itself: its relative complement is empty and hence open. There is no contradiction, because zero is not an available ambient limit in X. Likewise, the rational numbers are closed as a subset of themselves even though rational sequences can approach irrational numbers in the real line. Closedness never replaces a statement of the ambient space.

The closure of a set consists of points whose every positive-radius ball meets that set. In metric spaces these are exactly limits of sequences selected from the set, allowing constant sequences for points already present. Closure adds all missing accessible limits at once; it does not require a particular generating sequence. Open and closed are also not logical opposites: a set may be both, such as the whole space, or neither. In the discrete metric every set is both open and closed, because each point is isolated.