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Convergence of a sequence in a metric space

A sequence converges if points eventually lie arbitrarily close to the limit

Let (X,d)(X,d) be a . A sequence (xn)(x_n) in XX converges to a point a∈Xa\in X if

(∀ε>0)(∃N∈N)(∀n≥N): d(xn,a)<ε.(\forall \varepsilon>0)(\exists N\in\mathbb{N})(\forall n\ge N):\ d(x_n,a)<\varepsilon.

We write lim⁡n→∞xn=a\lim_{n\to\infty}x_n=a or xn→ax_n\to a.

Convergence in metric spaces is the foundation for defining and for studying .

Examples
  • In R\mathbb{R}, xn=1/nx_n=1/n converges to 00.
  • In any metric space, a constant sequence xn=xx_n=x converges to xx.
  • In the discrete metric, xn→ax_n\to a iff xn=ax_n=a for all sufficiently large nn.