Floor Function is Idempotent

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

Let $\left \lfloor{x}\right \rfloor$ be the floor of $x$ and $\left \lceil{x}\right \rceil$ be the ceiling of $x$.

Then the following apply:
 * $\left \lfloor{\left \lfloor{x}\right \rfloor}\right \rfloor = \left \lfloor{x}\right \rfloor$
 * $\left \lceil{\left \lceil{x}\right \rceil}\right \rceil = \left \lceil{x}\right \rceil$

That is, both the floor and ceiling functions are idempotent.

Proof
Let $y = \left \lfloor{x}\right \rfloor$.

By definition, $y$ is an integer.

Then from Integer Equals Floor And Ceiling, $\left \lfloor{y}\right \rfloor = y$.

So:
 * $\left \lfloor{\left \lfloor{x}\right \rfloor}\right \rfloor = \left \lfloor{x}\right \rfloor$

Similar for the ceiling function.