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I tried looking it up online but I have found nothing, therefore I am unsure whether what I am doing is right, I'd like to also know what I am doing wrong in the proof.

Remember that if two permutations are conjugate, they have the same parity. Consider the partition induced on $S_n$ by the conjugacy relation $S_n/C_{S_n}$. We know that the number of even permutations is exactly half the order of the $S_n$, hence if we consider $Q = \{[f_i]_{C_{S_n}}|f_i \text{ is even}\}$ we get that $|\bigcup Q|= |S_n \setminus (S_n \setminus A_n)| = \frac{|S_n|}{2} = |A_n|$. Consider now the partition $A_n/C_{A_n}$, we have that $[f_i]_{C_{A_n}} \subseteq [f_i]_{C_{S_n}} \in Q$, if there were an $x\in [f_i]_{C_{S_n}} \setminus [f_i]_{C_{A_n}}$, then we would get $|\bigcup Q|= \sum_{i} |[f_i]_{C_{S_n}}| > \sum_{i} |[f_i]_{C_{A_n}}| = |\bigcup A_n/C_{S_n}| = |A_n|$, hence a contradiction. Therefore it must be $[f_i]_{C_{A_n}} = [f_i]_{C_{S_n}}$.

I'm not convinced of the proof, yet I am not sure what I might be doing wrong. Thanks.

roxingby
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    Not necessarily: some conjugacy classes split in two. An easy example to see is $A_3$ in $S_3$: the conjugacy class of $(123)$ in $S_3$ is ${(123),(132)}$, but since $A_3$ is abelian the conjugacy class of $(123)$ is just ${(123)}$. – Arturo Magidin Apr 01 '22 at 22:14
  • Thanks for the example. But then, what is wrong with my proof? – roxingby Apr 01 '22 at 22:15
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    When you have a counterexample, just run your "proof" through the example and you will immediately spot the error. Your error is in the claim of inequality: the index sets are not the same, because you don't know ahead of time that you have the same number of equivalence classes. You may have multiple conjugacy classes of $A_n$ sitting inside a single conjugacy class in $S_n$. – Arturo Magidin Apr 01 '22 at 22:15
  • Oh, thanks a lot! – roxingby Apr 01 '22 at 22:17
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    By the way, I don't know what you tried to look up on-line, but I quickly found this post in this very site. – Arturo Magidin Apr 01 '22 at 22:18

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