Real Number is between Ceiling Functions

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

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

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

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.