Exercise 2, § 8

Let $G$ be the Hilbert space of sequences $(\lambda_1, \lambda_2, \ldots)$ of real numbers such that $\sum_i \lambda_i^2 < +\infty$. Consider $G$ as a real Lie group. Let $G_n$ be the set of $(\lambda_1, \lambda_2, \ldots) \in G$ such that $\lambda_m \in \frac{1}{m} \mathbf{Z}$ for $1 \leq m \leq n$. The $G_n$ are closed Lie subgroups of $G$ and hence $H = \bigcap_n G_n$ is a closed subgroup of $G$. But this group is totally disconnected and not discrete and hence is not a Lie subgroup of $G$.