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 And Ceiling, we have:
 * $$x = \left \lfloor {x} \right \rfloor \iff x \in \Z$$

Hence the result.

Also see

 * Ceiling of Floor is Floor