Dirichlet's unit theorem states that (i) the group of units, $\mathscr{U}_K$, of the ring of integers of a number field $K$ is finitely generated, and (ii) the rank of $\mathscr{U}_K$ is equal to $r_1 + r_2 - 1$, where $r_1$ is the number of real embeddings and $r_2$ the number of conjugate pairs of complex embeddings of $K$.

Q. What about a reference to an elementary, short proof of (i)?

In principle, (i) is much weaker than (ii), so the question shouldn't be so implausible. The reason why I'm asking is that I seem to have an alternative, extravagant proof of a well-known result, but for the approach to be of any potential interest I need (a reference to) an elementary, short proof of (i).