The proof in my lectures notes for the divergence of $a_n=n$ is as follows:
Let $L \in \mathbb{R}$. Let $\epsilon=1$. Let $N \in \mathbb{N}$. By the Archimedean property, $\exists n \in \mathbb{N}$ with $n > L$. Let $m=max(n,N)+1$. Then $m-L \geq n-L + 1$ and $m > N$. This shows L isn't a limit. I'm wondering why the proof didn't stop at the "$n > L$" part since it tells us that for whatever value $L$ we have we can find an $n$, which is obviously in the sequence, greater than L. And also, if the proof isn't supposed to end there, shouldn't the last step be "$m > n$" rather than "$m > N$?