Floor Function is Idempotent

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

Let $\floor x$ denote the floor of $x$.

Then:
 * $\floor {\floor x} = \floor x$

That is, the floor function is idempotent.

Proof
Let $y = \floor x$.

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

Then from Real Number is Integer iff equals Floor:
 * $\floor y = y$

So:
 * $\floor {\floor x} = \floor x$

Also see

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