Limit directions of a closed subgroup ($X_j\to 0,X_j/\Vert X_j\Vert \to X,\exp X_j\in H⟹\exp (tX)\in H$)

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 $X_j/\Vert X_j\Vert \to X$. For any real $t$, choose integers $m_j$ with $m_j\Vert X_j\Vert \to t$. Then $(\exp X_j{)}^{m_j}=\exp (m_j{X}_j)\to \exp (tX)$. 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.