Let $\displaystyle G=A^{\; *}_{H} A$ be the generalized free products amalgamating $\displaystyle H$. Let $\displaystyle G$ be residually finite. How to show that $\displaystyle A$ is $\displaystyle H$-separable?
I'm sure that $\displaystyle A$ must be $\displaystyle H$-separable. But I don't know how to show this.
I started with $\displaystyle x \in A \backslash H$.
Since $\displaystyle A$ is residually finite, then there exists $\displaystyle M \lhd_{f} A$ such that $\displaystyle x \notin M$.
I'm trying to prove by contradiction, by assuming $\displaystyle x \in HM$.
How can I continue from here?