(i) Prove that the ring of integers $\mathcal{O}_{K}$ in a real quadratic field $K$ contains a non-trivial unit. Any general results about lattices and convex bodies may be assumed.

(ii) State the general version of Dirichlet's unit theorem.

(iii) Show that for $K=\mathbb{Q}(\sqrt{7}), 8+3 \sqrt{7}$ is a fundamental unit in $\mathcal{O}_{K}$.

[You may not use results about continued fractions unless you prove them.]