Let $\beta \in S_n$ be an $r$-cycle. I am trying to show that $\beta \in A_n$ iff $r$ is odd.
By assumption we have $\beta = (a_1,a_2,a_3,...,a_r)$ for some $r \in \mathbb{N}$. Then assuming $\beta \in A_n$, it follows that $\beta$ is a product of an even number of transpositions and so that product must have $|r-1|$ products (an odd number). So $r$ is odd. If $r$ is odd then the cycle decomposition can be arranged $(a_1,a_r)(a_1,a_{r-1})(a_1,a_{r-2})...(a_1,a_2)$ so there are $r-1$ transpositions (an even number) implying that $\beta \in A_n$ since $r$ is assumed odd.
Is my reasoning sound?