[[Euclidean spaces]] is [[Complete metric spaces|complete]]. ###### Proof If $(x_{j})\in \mathbb{R}^n$ is Cauchy then for each $i\in \{ 1,2,3,\dots,n \}$ $x_{j,i}$ is also Cauchy. The [[General Principle of Convergence]] yields that $x_{j}$ is component-wise convergent and so by [[Equivalence of Component-wise Convergence and Convergence for Sequences in Euclidean Space]] $x_{j}$ is indeed convergent.