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?