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 $$k,l\ge N$$.

Note that Cauchy sequences are very closely related to convergence. Clearly, every convergent sequence is a Cauchy sequence, but the converse does not necessarily holds; it holds, by definition, for Banach spaces.

The advantage of Cauchy sequences compared to convergent sequences is that for Cauchy sequences we don't have to guess or to know to what the sequence convergences too (i.e., the $$v$$; see convergence). We just need to show that the norm difference shown above gets smaller and smaller. This is simpler to do Mathematically.