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:

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

$\blacksquare$


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