Ceiling of x+m over n/Proof 2

Proof
Let $f: \R \to \R$ be the real function defined as:
 * $\forall x \in \R: \map f x = \dfrac {x + m} n$

It is clear that $f$ is both strictly increasing and continuous on the whole of $\R$.

Let $\dfrac {x + m} n \in \Z$.

Then:

Thus:
 * $\forall x \in \R: \map f x \in \Z \implies x \in \Z$

So the conditions are fulfilled for McEliece's Theorem (Integer Functions) to be applied:
 * $\ceiling {\map f x} = \ceiling {\map f {\ceiling x} } \iff \paren {\map f x \in \Z \implies x \in \Z}$

Hence the result.