= Solution
Suppose that $H$ is not a <discrete subgroup>. The identity is then an accumulation point of nonidentity subgroup elements: translate any failure of isolation to the identity. Choose $h_j\in H\setminus\{e\}$ with $h_j\to e$, and use a <local exponential chart> to write $h_j=\exp X_j$, where $X_j\ne0$ and $X_j\to0$ in the real <Lie algebra> $\mathfrak g$.
Choose a <norm> on this finite-dimensional <vector space>. By <compactness> of its unit sphere, pass to a subsequence with $X_j/\|X_j\|\to X$, where $\|X\|=1$. For an arbitrary real $t$, choose an <integer> $m_j$ nearest to $t/\|X_j\|$. Then $m_j\|X_j\|\to t$ and
$$
h_j^{m_j}=\exp(m_jX_j)\longrightarrow\exp(tX).
$$
Every left-hand element belongs to $H$, including for negative $m_j$. Closedness gives $\exp(tX)\in H$. Since $t$ was arbitrary and $X\ne0$, the <limit directions of a closed subgroup> provide a nontrivial <one-parameter subgroup> in $H$. Its derivative at zero is $X$, so it cannot be the constant subgroup.
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