Floor of Non-Integer

Theorem
Let $x \in \R$ be a real number.

Let $x \notin \Z$.

Then:
 * $\left\lfloor{x}\right\rfloor < x$

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

Proof
From Floor is between Number and One Less:
 * $\left\lfloor{x}\right\rfloor \le x$

From Real Number is Integer iff equals Floor:
 * $x = \left \lfloor {x} \right \rfloor \iff x \in \Z$

But we have $x \notin \Z$.

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

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

Also see

 * Ceiling of Non-Integer