Proving that if $m \in \mathbb{Z}$, then $\left]m,m+1\right[ = \emptyset$ in $\mathbb{Z}$.

Question

Using the fact that $\mathbb{Z}$ is an ordered ring.

Is this the same as proving that $n>m \Leftrightarrow n \geq m+1$ (again, in $\mathbb{Z}$)? I take it that it is, but, I also have trouble proving that.

I'd appreciate any sort of guidance or hints.