Equivalence of Cauchy sequences/Proof

Statement

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:

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

And that this indeed actually defines an equivalence relation

Proof

Reflexivity

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Transitivity

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Symmetry

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