Floor is between Number and One Less

Theorem

 * $x - 1 < \floor x \le x$

where $\floor x$ denotes the floor of $x$.

Proof
By definition of floor function:
 * $\floor x \le x < \floor x + 1$

Thus by subtracting $1$:
 * $x - 1 < \paren {\floor x + 1} - 1 = \floor x$

So:
 * $\floor x \le x$

and:
 * $x - 1 < \floor x$

as required.