Cauchy sequence

A sequence $(a_n{)}_{n\ge 1}$ of real numbers is called a Cauchy sequence if its terms eventually become arbitrarily close to each other.

Definition

$(a_n{)}_{n\ge 1}$ is Cauchy if

$$\forall \epsilon >0\exists N\in \mathbb{N}\forall n,m>N:|a_n-a_m|<\epsilon$$

The difference to ordinary convergence is that we do not assume the limit in advance. We only require that the sequence becomes internally consistent.

Basic facts

Sequence is convergent in $\mathbb{R}$ $⟺$ Sequence is Cauchy

Example

The sequence $a_n=\frac{1}{n}$ is Cauchy because it converges to $0$.