When Are Discrete Subgroups Closed?

Let $H\subseteq G$ be a subgroup of a topological group $G$ (henceforth abbreviated “group”). If the induced topology on $H$ is discrete, then we say that $H$ is a discrete subgroup of $G$. A commonplace example is the subgroup $\mathbb{Z}\subseteq \mathbb{R}$: the integers are normal subgroup of the real numbers (with the standard topology).

Observe that in $\mathbb{R}$, the subset $\mathbb{Z}$ is also a closed set. What about an arbitrary discrete subgroup $H\subseteq\mathbb{R}$? In other words, if $H$ is discrete, is $H$ necessarily closed?
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