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