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.

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