# Limit of a sequence

From Citizendium, the Citizens' Compendium

The mathematical concept of **limit of a sequence** provides a rigorous definition of the idea of a sequence converging towards a point called the limit.

Suppose *x*_{1}, *x*_{2}, ... is a sequence of real numbers.
We say that the real number *L* is the *limit* of this sequence and we write

if and only if for every real number ε > 0 there exists a natural number *n*_{0} such that for all *n* > *n*_{0} we have |*x*_{n} − *L*| < ε. The number *n*_{0} will in general depend on ε.