Conjugacy-separable implies characters of finite-dimensional representations over complex numbers separate conjugacy classes

Statement
Suppose $$G$$ is a fact about::conjugacy-separable group, i.e., any two distinct conjugacy classes can be separated in a finite quotient group. Then, given any two distinct conjugacy classes, there is a finite-dimensional representation of $$G$$ over the field of complex numbers $$\mathbb{C}$$ such that the character value of the representation at the two conjugacy classes is different.

In fact, instead of taking $$\mathbb{C}$$, we can simply take the cyclotomic algebraic closure of the rational numbers, i.e., the field obtained by adjoining all roots of unity to the rational numbers.

Facts used

 * 1) uses::Sufficiently large implies splitting: This says that a field that contains all primitive $$m^{th}$$ roots of unity where $$m$$ is the exponent of the group is a splitting field.
 * 2) uses::Splitting implies characters separate conjugacy classes