Cauchy sequence

A sequence $\{x_n \}_{n\in \mathbb N}$ in a metric space $(X,d)$ is said to be a Cauchy sequence if and only if the following is true: for every real number $\varepsilon > 0$ there exists $N \in \N$ such that $N\le m,n\in\N$ implies that $d(x_m,x_n)<\varepsilon$ .

If we consider $\R$ as a metric space with the distance between $x, y \in \R$ defined as the usual Euclidean distance $|x-y|$ , then a sequence $\{x_n \}_{n\in \mathbb N}$ of real numbers is Cauchy if and only if the following is true: for every real number $\varepsilon > 0$ there exists $N \in \N$ such that $N\le m,n\in\N$ implies that $|x_m-x_n|<\varepsilon$ .

In every metric space, every convergent sequence is a Cauchy sequence. A metric space in which every Cauchy sequence converges is said to be complete. One may show that all compact metric spaces and all Euclidean spaces are complete. Furthermore, closed subsets of complete metric spaces are complete.