Floor Function is Idempotent

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

Let $\left \lfloor{x}\right \rfloor$ be the floor of $x$.

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

That is, the floor function is idempotent.

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

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

Then from Real Number is Integer iff equals Floor:
 * $\left \lfloor{y}\right \rfloor = y$

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

Also see

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