Cauchy Criterion in Rk

Cauchy criterion in $\mathbb{R}^k$: Let $(x_n)$ be a sequence in $\mathbb{R}^k$, with distance measured by the Euclidean norm (equivalently, the Euclidean metric). Then $(x_n)$ converges in $\mathbb{R}^k$ if and only if it is a Cauchy sequence , meaning: for every $\varepsilon>0$ there exists $N$ such that for all $m,n\ge N$, $\|x_n-x_m\|<\varepsilon$.

This is the completeness of Euclidean space, i.e. that $\mathbb{R}^k$ is a complete metric space . In one dimension it is one of the standard consequences of the completeness axiom .