Cauchy sequence

A sequence $$\{\mathbf{v}_k\}_{k=1}^{\infty}$$ in a normed vector space $$V$$ is called a Cauchy sequence if (Def. 3.1.1 ) for any $$\epsilon>0$$, there exists $$N\in \mathbb{N}$$ such that

$$||\mathbf{v}_k-\mathbf{v}_l||<\epsilon$$ whenever $$k,l\ge N$$.