Ceiling of Floor is Floor

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

That is, the ceiling of the floor is the floor.

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

By Floor Function is Integer, we have that $y \in \Z$.

Then from Real Number is Integer iff equals Ceiling, we have:
 * $\left \lceil{y}\right \rceil = y$

So:


 * $\left \lceil {\left \lfloor {x}\right \rfloor}\right \rceil = \left \lfloor {x}\right \rfloor$

Also see

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