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 definition of the floor function, we have that $\left \lfloor {x} \right \rfloor \in \Z$.

From Integer equals Floor and Ceiling, we have:
 * $x = \left \lceil {x} \right \rceil \iff x \in \Z$

Hence the result.

Also see

 * Floor of Ceiling is Ceiling