Ceiling of Floor is Floor

Theorem

Let $x \in \R$ be a real number.

Let $\floor x$ denote the floor of $x$, and $\ceiling x$ denote the ceiling of $x$.

Then:

$\ceiling {\floor x} = \floor x$

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

Proof

Let $y = \floor x$.

By Floor Function is Integer, we have that $y \in \Z$.

Then from Real Number is Integer iff equals Ceiling, we have:

$\ceiling y = y$

So:

$\ceiling {\floor x} = \floor x$

$\blacksquare$

Also see

• Floor of Ceiling is Ceiling
• Floor Function is Idempotent
• Ceiling Function is Idempotent

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $2$