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

Statement

Suppose $G$ is a 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. 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. Splitting implies characters separate conjugacy classes

Proof

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