Ceiling minus Real Number

Theorem

$\forall x \in \R: \ceiling x - x \in \hointr 0 1$

where $\ceiling x$ denotes the ceiling of $x$.

$\ds \ceiling x - 1$

$<$

$\ds x \le \ceiling x$

$\ds \leadsto \ \ $

$\ds \ceiling x - 1 - \ceiling x$

$<$

$\ds x - \ceiling x \le \ceiling x - \ceiling x$

$\ds \leadsto \ \ $

$\ds -1$

$<$

$\ds x - \ceiling x \le 0$

$\ds \leadsto \ \ $

$\ds 1$

$>$

$\ds \ceiling x - x \ge 0$

$\ds \leadsto \ \ $

$\ds \ceiling x - x$

$\in$

$\ds \hointr 0 1$

$\blacksquare$