# Proving something is not a Cauchy sequence (Theory Proof)

I need to prove that $${\{X_n\}}$$ is not a Cauchy sequence. I understand that in order to prove this, I need to prove that $$\exists\ \epsilon\gt0\ | \forall N \in \Bbb N,$$ if, there is $$n,m \geq N$$, then $$|X_n-X_m|\geq \epsilon$$

So my question, what happens if $$n=m$$?, then $$|X_n-X_m|=0$$ and therefore $$|X_n-X_m|<\epsilon$$ because $$\epsilon \gt 0$$ and it would never hold, because for $$\forall N$$ there would exist an $$n,m$$ that satisfy this. So, do the $$n,m$$ need to be different?

Thanks for any help! (:

To assert that $$(X_n)_{n\in\mathbb N}$$ is not a Cauchy sequence means that$$(\exists\varepsilon>0)(\forall N\in\mathbb N)(\exists m,n\in\mathbb N):m,n\geqslant N\text{ and }\lvert X_m-X_n\rvert\geqslant\varepsilon.$$So, you don't need to worry about the $$m=n$$ possibility. Given $$N\in\mathbb N$$, you only need to prove that there is some $$m\geqslant N$$ and there is some $$n\geqslant N$$ such that $$\lvert X_m-X_n\rvert\geqslant\varepsilon$$. And, of course you will not pick $$m=n$$.
We are not saying that $$|X_n-X_m| \geq \epsilon$$ whenever $$n,m \geq N$$. We are only saying that this holds for some $$n,m \geq N$$, Surely such $$n$$ and $$m$$ cannot be equal.
A little knowledge of quantifiers may help you here. The negation of universal quantifier '$$\lor$$' (for all)is existential quantifier '$$\exists$$' (for some). Now just negate the Cauchy's criterian of existence of limits to get the one you needed.