Equivalence of Cauchy sequences/Definition

This sub-page is ideal for transclusion, where ever a reminder of the definition of equivalence of Cauchy sequences is required.

Definition

Given two Cauchy sequences, $(a_n)_{n=1}^\infty$ and $(b_n)_{n=1}^\infty$ in a metric space $(X,d)$ we define them as equivalent if:

• $$\forall\epsilon>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n>N\implies d(a_n,b_n)<\epsilon]$$