Floor is between Number and One Less

Theorem

 * $x - 1 < \left\lfloor{x}\right\rfloor \le x$

where $\left\lfloor{x}\right\rfloor$ is the floor of $x$.

Proof
By definition of floor function:
 * $\left\lfloor{x}\right\rfloor \le x < \left\lfloor{x}\right\rfloor + 1$

Thus by subtracting $1$:
 * $x - 1 < \left({\left\lfloor{x}\right\rfloor + 1}\right) - 1 = \left\lfloor{x}\right\rfloor$

So:
 * $\left\lfloor{x}\right\rfloor \le x$

and:
 * $x - 1 < \left\lfloor{x}\right\rfloor$

as required.