Ceiling Function is Idempotent

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

Let $\left \lceil{x}\right \rceil$ denote the ceiling of $x$.

Then:
 * $\left \lceil{\left \lceil{x}\right \rceil}\right \rceil = \left \lceil{x}\right \rceil$

That is, the ceiling function is idempotent.

Proof
Let $y = \left \lceil{x}\right \rceil$.

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

Then from Real Number is Integer iff equals Ceiling:
 * $\left \lceil{x}\right \rceil = y$

So:
 * $\left \lceil{\left \lceil{x}\right \rceil}\right \rceil = \left \lceil{x}\right \rceil$

Also see

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