Let $A$ be a C$^{*}$-algebra. Let $\operatorname{Aut}(A)$ denote the set of all $*$-isomorphisms from $A$ onto itself. We call an element $\alpha \in \operatorname{Aut}(A)$ an inner automorphism if there is a unitary element $u$ in the unitization $\widetilde{A}$ of $A$ such that $\alpha(a)=uau^{*}$ for all $a$ in $A$.
An $\alpha\in \operatorname{Aut}(A)$ is called approximately inner if for every finite subset $F$ of $A$ and every $\epsilon>0$, there is an inner automorphism $\beta$ such that $\|\alpha(a)-\beta(a)\|<\epsilon$. We denote the set of approximately inner automorphisms of $A$ be $\overline{\operatorname{Inn}}(A)$.
I am trying to prove that $\overline{\operatorname{Inn}}(A)$ is a normal subgroup of $\operatorname{Aut}(A)$.
I know that the set of all inner automorphisms of $A$ form a normal subgroup of $\operatorname{Aut}(A)$, and using this I was able to show that $\overline{\operatorname{Inn}}(A)$ is closed under composition and that it is a normal subgroup assuming it is a group.
I have been struggling to show that if $\alpha\in\overline{\operatorname{Inn}}(A)$, then $\alpha^{-1}\in\overline{\operatorname{Inn}}(A)$.
This is all I need to complete the proof. I am having trouble determining what inner automorphism to approximate $\alpha^{-1}$ with for a given $F$ and $\epsilon$. Any help is very much appreciated.
Thank you.