Real Number is between Ceiling Functions

Theorem

 * $\forall x \in \R: \left \lceil {x - 1} \right \rceil \le x < \left \lceil {x} \right \rceil$

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

Proof
$\left \lceil {x} \right \rceil$ is defined as:


 * $\left \lceil {x} \right \rceil = \inf \left({\left\{{m \in \Z: m \ge x}\right\}}\right)$

So $\left \lceil {x} \right \rceil \ge x$ by definition.

Now $\left \lceil {x - 1} \right \rceil < \left \lceil {x} \right \rceil$, so by the definition of the infimum:
 * $\left \lceil {x - 1} \right \rceil > x$

The result follows.