Ordering/Examples/Integer Difference on Reals

Example of Ordering
Let $\preccurlyeq$ denote the relation on the set of real numbers $\R$ defined as:
 * $a \preccurlyeq b$ $b - a$ is a non-negative integer

Then $\preccurlyeq$ is an ordering on $\R$.

Reflexivity
We have that:


 * $\forall a \in \R: a - a = 0 \in \Z_{\ge 0}$

Thus:
 * $\forall a \in \R: a \preccurlyeq a$

So $\preccurlyeq$ has been shown to be reflexive.

Transitivity
Let $a, b, c \in \R$ such that:

So $\preccurlyeq$ has been shown to be transitive.

Antisymmetry
Let $a, b \in \R$ such that:

So $\preccurlyeq$ has been shown to be antisymmetric.

$\preccurlyeq$ has been shown to be reflexive, transitive and antisymmetric.

Hence by definition it is an ordering.