Real Number 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.

$\blacksquare$