# Equivalence of Cauchy sequences/Definition

From Maths

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, [ilmath](a_n)_{n=1}^\infty[/ilmath] and [ilmath](b_n)_{n=1}^\infty[/ilmath] in a metric space [ilmath](X,d)[/ilmath] we define them as equivalent if^{[1]}:

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