 Solution

The diagonal mapis 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 ${\mathbb{I}}_K$.