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:

$\floor {\floor x} = \floor x$

$\blacksquare$

Also see

• Ceiling Function is Idempotent
• Floor of Ceiling is Ceiling
• Ceiling of Floor is Floor