Ceiling of x+m over n/Proof 2

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

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:
 * $\forall x \in \R: f \left({x}\right) \in \Z \implies x \in \Z$

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

Hence the result.