Knowlpedia

Limit of a sequence

A point x such that x_n becomes arbitrarily close to x as n→∞.

Let (X,d)(X,d) be a and let (xn)(x_n) be a sequence in XX. A point x∈Xx\in X is the limit of (xn)(x_n) if

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

One writes x=lim⁡n→∞xnx=\lim_{n\to\infty}x_n or xn→xx_n\to x.

The limit (when it exists) summarizes the eventual behavior of a sequence. In metric spaces, a sequence has at most one limit (see ).

Examples
  • lim⁡n→∞(1/n)=0\lim_{n\to\infty} (1/n)=0 in R\mathbb{R}.
  • If xn=2x_n=2 for all nn, then lim⁡n→∞xn=2\lim_{n\to\infty}x_n=2.
  • The sequence (−1)n(-1)^n has no limit in R\mathbb{R}.