Ceiling Function is Idempotent
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Theorem
Let $x \in \R$ be a real number.
Let $\ceiling x$ denote the ceiling of $x$.
Then:
- $\ceiling {\ceiling x} = \ceiling x$
That is, the ceiling function is idempotent.
Proof
Let $y = \ceiling x$.
By Ceiling Function is Integer, $y$ is an integer.
Then from Real Number is Integer iff equals Ceiling:
- $\ceiling y = y$
So:
- $\ceiling {\ceiling x} = \ceiling x$
$\blacksquare$