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
By Floor Function is Integer, we have that $\left \lfloor {x} \right \rfloor \in \Z$.

From Real Number is Integer iff equals Ceiling, we have:
 * $\left \lceil {\left \lfloor {x}\right \rfloor}\right \rceil = \left \lfloor {x}\right \rfloor$

Also see

 * Floor of Ceiling is Ceiling
 * Floor of Floor is Floor
 * Ceiling of Ceiling is Ceiling