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.