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