Q:

How to prove that the adjoint group is a Lie subgroup of $Gl(\mathfrak{g})$

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. Let $Ad: G \rightarrow GL(\mathfrak{g})$ be the Adjoint representation. I want to prove that $Ad(G)$ is a Lie subgroup of $GL(\mathfrak{g})$.
Here is what I tried: From Cartan's Theorem, it suffices to show that $Ag(G)$ is closed in $GL(\mathfrak{g})$. So take $x \in Ad(G)$ and $(Ad(g_n))_n$ a sequence in $Ad(G)$ that converges to $x$, with $g_n \in G$. I can't continue from here. It may be possible that some additional other hypothesis are required. 
Thanks!

A:

User proposition 7.1 in San Martin book. http://www.ime.unicamp.br/~smartin/cursos/grupolie-2013/gruplie0.pdf

