Math 2.0

General topology Definition

Convergence of a sequence in a metric space

Builds on Metric space

Definition

Let (X,d)(X, d) be a metric space, let (xn)n≥1(x_n)_{n \ge 1} be a sequence of points of XX, and let x∈Xx \in X. The sequence (xn)(x_n) converges to xx when for each real ε>0\varepsilon > 0 there is an integer NN such that d(xn,x)<εd(x_n, x) < \varepsilon for all n≥Nn \ge N. Then xx is a limit of (xn)(x_n), with the notation xn→xx_n \to x or lim⁡n→∞xn=x\lim_{n \to \infty} x_n = x.

Remarks

Equivalently, xn→xx_n \to x if and only if d(xn,x)→0d(x_n, x) \to 0 in ℝ\mathbb{R}. A sequence in a metric space has at most one limit. This follows from the triangle inequality, and also from the fact that limits are unique in Hausdorff spaces. For X=ℝX = \mathbb{R} with the usual metric, this is the usual limit of a real sequence.

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  16. Convergence of a sequence in a metric space definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Limit_of_a_sequence, Metric spaces
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