Given any x R prove that there exists a unique n Z such that n - 1 x < n.
Proof: Suppose x R. By the Archimedean property, there is an n Z such that n > x. Since N is well-defined, we have that n - 1 N and n - 1 < n. Then it follows that n - 1 x < n for x 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?