Integers are not Densely Ordered

Theorem
The integers $\Z$ are not densely ordered.

That is:
 * $\forall n \in \Z: \not \exists m \in \Z: n < m < n + 1$

Proof
By definition of immediate successor element, this is equivalent to the statement:
 * $\forall n \in \Z: n + 1$ is the immediate successor to $n$

We have that Integers form Ordered Integral Domain.

From One Succeeds Zero in Well-Ordered Integral Domain:
 * $\not \exists r \in \Z: 0 < r < 1$

From Properties of Ordered Ring:
 * $a < b \implies n + a < n + b$

Putting $a = 0, b = 1, m = n + r$:
 * $\not \exists m \in \Z: n + 0 < m < n + 1$

Hence the result.