How to define the negation of "Cauchy sequence"?

Question

For the given sequences we define what a Cauchy sequence is.

But my problem is: How do we define "not a Cauchy sequence"?

Answer 1

Cauchy sequence: $\forall\epsilon>0\exists N\in\mathbb{N}:\forall (n,m\geq N\implies |x_m-x_n|<\epsilon)$

Negation: $\exists\epsilon>0\forall N\in\mathbb{N}:\exists n,m\geq N\wedge |x_m-x_n|\geq \epsilon$