Suppose that there exists some $N \in \mathbb{Z}$ which has the property $N \geq n \forall n \in \mathbb{Z}$. This means that $N$ is at least as large as any integer.

Now consider that for any $n \in \mathbb{Z}$, we have $(n + 1) \in \mathbb{Z}$ as well. Also, since $0 \lt 1$, we have $(0 + N) \lt (1 + N)$ so that $N \lt (N + 1)$. Thus, we have constructed an integer, $(N + 1)$ which is greater than $N$. Since $N$ was assumed to be the greatest integer, we have shown by contradiction that such a property cannot hold. Thus, there is no largest integer.