Ceiling is between Number and One More

Theorem

 * $x \le \ceiling x < x + 1$

where $\ceiling x$ is the ceiling of $x$.

Proof
From Number is between Ceiling and One Less:
 * $\ceiling x - 1 < x \le \ceiling x$

Thus by adding $1$:
 * $x + 1 > \paren {\ceiling x - 1} + 1 = \ceiling x$

So:
 * $x \le \ceiling x$

and:
 * $\ceiling x < x + 1$

as required.