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.

$\blacksquare$

Also see

• Floor of Ceiling is Ceiling

Sources

• Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (1968): $\S 1.2.4$: Exercise $2$