$\iota\left(\mathbb Q\right)\subset\mathbb A_\mathbb Q$ is a lattice (diagonal embedding).

Question

How could I show that the diagonal embedding of $\mathbb Q$ into $\mathbb A_\mathbb Q$ is a lattice?

Here, $\mathbb A_\mathbb Q$ is the set of adeles, and $\iota:\mathbb Q\to\mathbb A_\mathbb Q$ is defined by $q\mapsto\left(q,q,\dots\right)$.

Answer 1

Hint To show that $\iota(\mathbb Q)$ is discrete in $\mathbb A_\mathbb Q$ you need to show that if $\iota(q_n) \to 0$ then $q_n$ is eventually 0. Just look at what does this mean in each component.

For relatively denseness, show that $$\iota(\mathbb Q) + [0,1] \times \prod_p \mathbb Z_p =\mathbb A_\mathbb Q$$