Not Cauchy Proof By Definition

Question

How do I prove that $\{n\}$ is not Cauchy?

So far, this is what I have so far:

Let $ε=0.$ Let N ∈ ℕ be given. Assume $n,m ≥ N$ and $|X_{n+1}-X_n|=|N+1-N|=1>\epsilon .$ Thus $\{n\}$ is not Cauchy.

Answer 1

Let $\varepsilon=1$. Fix $p\in\mathbb N$. Then, if you take $m=p$ and $n=p+1$, then $m,n\geqslant p$ and $|x_m-x_n|=1\geqslant\varepsilon$.

What you did is wrong, because it is part of the definition of Cauchy sequence that $\varepsilon>0$.