Floor of Ceiling is Ceiling

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

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

Proof
By definition of the ceiling function, we have that $\left \lceil {x} \right \rceil \in \Z$.

From Integer equals Floor, we have:
 * $x = \left \lfloor {x} \right \rfloor \iff x \in \Z$

Hence the result.

Also see

 * Ceiling of Floor is Floor