Limit of a sequence

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 x₁, x₂, ... is a sequence of real numbers. We say that the real number L is the limit of this sequence and we write

$\lim _{n\to \infty }x_{n}=L$

if and only if for every real number ε > 0 there exists a natural number n₀ such that for all n > n₀ we have |x_n − L| < ε. The number n₀ will in general depend on ε.