Ceiling Function is Idempotent

Theorem
Let $x \in \R$ be a real number.

Let $\ceiling x$ denote the ceiling of $x$.

Then:
 * $\ceiling {\ceiling x} = \ceiling x$

That is, the ceiling function is idempotent.

Proof
Let $y = \ceiling x$.

By Ceiling Function is Integer, $y$ is an integer.

Then from Real Number is Integer iff equals Ceiling:
 * $\ceiling y = y$

So:
 * $\ceiling {\ceiling x} = \ceiling x$

Also see

 * Floor Function is Idempotent
 * Floor of Ceiling is Ceiling
 * Ceiling of Floor is Floor