Definition (Cauchy Sequence)

Let $(V,\|\cdot\|)$ be a normed vector space. A sequence $(v_{i})_{i\in\mathbb{N}}$ is a Cauchy sequence if $$\forall\epsilon\in\mathbb{R}^{+},\exists N\in\mathbb{N}\text{ s.t. }\lVert v_{k}-v_{i} \rVert <\epsilon,\forall j,k>N$$i.e., for every $\epsilon\in\mathbb{R}^{+}$, $\exists N\in\mathbb{N}$ such that $\|v_{k}-v_{i}\|<\epsilon$ for every $j,k>N$.