= Limit directions of a closed subgroup
{title2=$X_j\to0,\quad X_j/\|X_j\|\to X,\quad\exp X_j\in H\quad\Longrightarrow\quad\exp(tX)\in H$}
Let $H$ be a closed <subgroup> of a <Lie group> $G$, and let nonzero vectors $X_j$ in the ambient <Lie algebra> $\mathfrak g$ tend to zero with $\exp X_j\in H$ and $X_j/\|X_j\|\to X$. For any real $t$, choose <integers> $m_j$ with $m_j\|X_j\|\to t$. Then $(\exp X_j)^{m_j}=\exp(m_jX_j)\to\exp(tX)$. Closedness gives $\exp(tX)\in H$. <Compactness> of the unit sphere supplies a limit direction from every sequence approaching the identity through nonidentity subgroup elements.
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