# Floor of Ceiling is Ceiling

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## 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

Let $y = \left \lceil{x}\right \rceil$.

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

From Real Number is Integer iff equals Floor, we have:

$\left \lfloor{y} \right \rfloor = y$

So:

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

$\blacksquare$