Given anyxRprove that there exists a uniquenZsuch that n - 1 x < n.

Proof:SupposexR. By the Archimedean property, there is annZsuch thatn>x.SinceNis well-defined, we have thatn -1Nandn -1 <n. Then it follows thatn -1x<nforxR.

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?