How do we know that $\mathbb{Z} + \sqrt{2}\mathbb{Z}$ isn't closed?

Question

The Minkowski sum of closed sets needn't be closed; $\mathbb{Z} + \sqrt{2}\mathbb{Z}$ is the canonical example. However, its not clear to me how to prove this.

Question. How can we prove that $\mathbb{Z} + \sqrt{2}\mathbb{Z}$ isn't closed?

Answer 1

$\mathbb{Z}+\sqrt{2}\mathbb{Z}$ is a subgroup of $(\mathbb{R},+)$, as any subgroup of $\mathbb{R}$ is dense or mono-gene (generated by one element), and it is easy to show that it is not mono-gene, hence dense, so not closed because it is $\neq \mathbb{R}$.