Tangent space of closure of subgroup of Lie group I am working my way through Stillwell's Naive Lie Theory.  I am looking for suggestions how to address question 8.2.4 - Show that 
$\{\text{sequential tangents to }H\} = T_{\mathbf 1}\overline H$ where $H$ is an arbitrary subgroup of a matrix Lie group?

If $X$ is a sequential tangent vector to $H$ at $\mathbf 1$, then it is also a sequential tangent vector to $\overline H$ at $\mathbf 1$ and therefore $X\in T_{\mathbf 1}\overline H$. (In Stillwell's book, this is proved right after the definition of sequential tangent vector.)
Now, let $X\in T_{\mathbf 1}\overline H$. Then there is a sequence $(A_n)_{n\in\mathbb N}$ of points of $\overline H$ such that$$X=\lim_{n\to\infty}n(A_n-\mathbf{1}).$$For each $n\in\mathbb N$, let $X_n\in H$ be such that $\|A_n-X_n\|\leqslant\frac1{n^2}$. Then\begin{align}X&=\lim_{n\to\infty}n(A_n-\mathbf{1})\\&=\lim_{n\to\infty}n(X_n-\mathbf{1})+\lim_{n\to\infty}n(A_n-X_n)\\&=\lim_{n\to\infty}n(X_n-\mathbf{1})\\&=\lim_{n\to\infty}\frac{X_n-\mathbf{1}}{1/n}\end{align}and therefore $X$ is a sequential tangent vector to $H$.
