Cauchy Sequence

Definition

A Cauchy sequence is a Sequence whose elements become arbitrarily close to each other as the sequence progresses. $(a_n)\text{s.t.}\forall ϵ>0,\exists N\in \mathbb{N}\text{s.t.}(\forall m,n\in \mathbb{N}\text{s.t.}m,n>N,|a_m-a_n|<ϵ)$

Facts

Every convergent sequence is a Cauchy sequence

In ${\mathbb{R}}^n$ space, Cauchy sequence converges.