Proof involving the Archimedean property.

Given any *x* $\displaystyle \in$ **R** prove that there exists a unique *n* $\displaystyle \in$ **Z** such that n - 1 $\displaystyle \leq$ x < n.

**Proof: **Suppose *x *$\displaystyle \in$ **R**. By the Archimedean property, there is an *n *$\displaystyle \in$ **Z** such that *n *> *x. *Since **N** is well-defined, we have that *n - *1 $\displaystyle \in$ **N **and *n -* 1 < *n*. Then it follows that *n - *1 $\displaystyle \leq$ *x *< *n *for *x *$\displaystyle \in$ **R.**

I feel like I've jumped to conclusions, but to my non-math-genius brain, I feel like this is correct. Can someone prod me along?

(Clapping)