Cauchy criterion for convergence in R^k

Cauchy criterion for convergence in $\mathbb{R}^k$: A sequence $(x_n)$ in $\mathbb{R}^k$ converges (with respect to the Euclidean norm ) if and only if it is a Cauchy sequence ; i.e., $x_n\to x \text{ for some } x\in\mathbb{R}^k \quad\Longleftrightarrow\quad \forall \varepsilon>0\;\exists N\;\forall m,n\ge N:\ \|x_n-x_m\|<\varepsilon.$

This is the completeness of Euclidean space and is central to analysis: it allows one to prove convergence by controlling pairwise distances rather than guessing the limit.

Proof sketch (optional): If $x_n\to x$ then $\|x_n-x_m\|\le \|x_n-x\|+\|x_m-x\|$ shows Cauchy. Conversely, if $(x_n)$ is Cauchy, each coordinate sequence is Cauchy in $\mathbb{R}$ and hence convergent; the vector of coordinate limits is the limit in $\mathbb{R}^k$.