## In a real algebraic integer ring containing a nontrivial unit, there exist units of arbitrarily small absolute value

Let $K$ be an algebraic number field consisting only of real numbers with ring of integers $R$, and suppose there exists a nontrivial unit $\zeta \in R$ (i.e. $\zeta \neq \pm 1$). Show that there exist units in $R$ having arbitrarily small absolute value.

Suppose (without loss of generality) that $\zeta > 0$. Suppose also (without loss of generality) that $\zeta < 1$. (Otherwise take $-\zeta$ or $\zeta^{-1}$ as needed.) Then $|\zeta^k|$ can be made arbitrarily small.