Number less than Integer iff Floor less than Integer

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

Let $\left \lfloor{x}\right \rfloor$ denote the floor of $x$.

Let $n \in \Z$ be an integer.

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

Necessary Condition
Let $x < n$.

By definition of the floor of $x$:
 * $\left \lfloor {x} \right \rfloor \le x$

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

Sufficient Condition
Let $\left \lfloor{x}\right \rfloor < n$.

We have that:
 * $\forall m, n \in \Z: m < n \iff m + 1 \le n$

and so:
 * $(1): \quad \left \lfloor{x}\right \rfloor + 1 \le n$

Then:

Hence the result:
 * $\left \lfloor{x}\right \rfloor < n \iff x < n$