# 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.

$\ceiling y = y$

So:

$\ceiling {\ceiling x} = \ceiling x$

$\blacksquare$