Number is between Ceiling and One Less

Theorem

$\ceiling x - 1 < x \le \ceiling x$

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

Proof

By definition of ceiling of $x$:

$\forall x \in \R: \ceiling x = \map \inf {\set {m \in \Z: m \ge x} }$

By definition of infimum:

$\ceiling x \ge x$

Also by definition of infimum:

$\ceiling x - 1 \not \ge x$

as $\ceiling x$ is the smallest integer with that property.

That is:

$x > \ceiling x - 1$

Hence the result.

$\blacksquare$