Number is between Ceiling and One Less

Theorem

 * $\left \lceil {x} \right \rceil - 1 < x \le \left \lceil {x} \right \rceil$

where $\left \lceil {x} \right \rceil$ denotes the ceiling of $x$.

Proof
By definition of ceiling of $x$:
 * $\forall x \in \R: \left \lceil {x} \right \rceil = \inf \left({\left\{{m \in \Z: m \ge x}\right\}}\right)$

By definition of infimum:
 * $\left \lceil {x} \right \rceil \ge x$

Also by definition of infimum:
 * $\left \lceil {x} \right \rceil - 1 \not \ge x$

as $\left \lceil {x} \right \rceil$ is the smallest integer with that property.

That is:
 * $x > \left \lceil {x} \right \rceil - 1$

Hence the result.