= Solution
The diagonal map
$$
K^\times\longrightarrow\mathbb I_K,
\qquad
\alpha\longmapsto(\alpha)_v,
$$
is an injective homomorphism because every completion map $K\to K_v$ is injective. The diagonal copy of $K$ is discrete in the <adele ring>. Choose an adelic neighbourhood $W$ of $1$ with $W\cap K=\{1\}$. By the subspace description of the idele topology, $W\cap\mathbb I_K$ is an idele neighbourhood of $1$ meeting diagonal $K^\times$ only at $1$. Translation proves that $K^\times$ is a discrete subgroup of $\mathbb I_K$.
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