Integers are not Densely Ordered
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- $\forall n \in \Z: \not \exists m \in \Z: n < m < n + 1$
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.
- $\not \exists r \in \Z: 0 < r < 1$
- $a < b \implies n + a < n + b$
Putting $a = 0, b = 1, m = n + r$:
- $\not \exists m \in \Z: n + 0 < m < n + 1$
Hence the result.