Real Number is between Ceiling Functions

Theorem

$\forall x \in \R: \ceiling {x - 1} \le x < \ceiling x$

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

Proof

$\ceiling x$ is defined as:

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

So $\ceiling x \ge x$ by definition.

Now $\ceiling {x - 1} < \ceiling x$, so by the definition of the infimum:

$\ceiling {x - 1} > x$

The result follows.

$\blacksquare$