Discrete subrings of $\mathbb R$

Question

Describe all discrete unital subrings of $\mathbb R$.

'Discrete subrings' = 'subrings which are discrete sets'

My attempt is as follows. The conjecture is that the only such subring is $\mathbb Z$. Let $R$ be a discrete subring of $\mathbb R$. Clearly, $\mathbb Z\subseteq R$ since $1\in R$ and hence $\pm n\in R$ for any integer $n$. It remains to show that $R$ contains nothing else than $\mathbb Z$.

I'm stuck at that point. Any suggestions/hints?

Answer 1

Hint If $R$ contains some $\alpha \notin \mathbb Z$, then by subtracting the integral part, $R$ contains some $0< \beta <1$.

Then $R$ contains $\beta^n$.