$\Psi(\operatorname{Aut}(G))$ is a closed subgrp of $\operatorname{Aut}(\mathfrak g)$ in the construction of a Lie grp structure of Aut$(G)$

Question

To construct a Lie group structure on $\operatorname{Aut}(G)$ of a connected Lie group $G$ with Lie algebra $\mathfrak g$ the following steps are followed:

1. $\Psi:\operatorname{Aut}(G)\to \operatorname{Aut}(\mathfrak g): f \to T_ef$ is proven to be an injective group homom.

2. Show that if $(T_e(f_n))_{n=1}^{\infty}$ converges in $\operatorname{Aut}(\mathfrak g)$ , then $(f_n)_{n=1}^{\infty}$ is a sequence in $\operatorname{Aut}(G)$, such that $\lim_{n\to \infty} f_n$ exists as an continuous smooth automorphism of the abstract group $G$

3. Observe that $\Psi(\operatorname{Aut}(G))$ is a closed subgroup of $\operatorname{Aut}(\mathfrak g)$

Assuming 1 and 2 are proven, How do I show step 3?