Conjugacy-separable implies residually finite

Statement
Any conjugacy-separable group is a residually finite group.

Conjugacy-separable group
A group $$G$$ is termed conjugacy-separable if given any elements $$x,y \in G$$ that are not conjugate in $$G$$, there exists a normal subgroup of finite index $$N$$ such that the images of x and $$y$$ in $$G/N$$ are not conjugate.

Residually finite group
A group $$G$$ is termed residually finite if given any non-identity element $$x \in G$$, there is a normal subgroup of finite index $$N$$ in $$G$$ such that $$x$$ is not in $$N$$.

Proof
Given: A conjugacy-separable group $$G$$, a non-identity element $$x \in G$$.

To prove: There is a normal subgroup of finite index $$N$$ in $$G$$ such that $$x$$ is not in $$N$$.

Proof: Let $$e$$ be the identity element. Since no non-identity element is conjugate to the identity element, $$x$$ and $$e$$ are in distinct conjugacy classes. Thus, there exists a normal subgroup $$N$$ of finite index such that the image of $$x$$ is not conjugate to the image of $$e$$. In particular, this implies that $$x$$ is not contained in $$N$$, so we are done.