Solution: We recall that a transposition is a $2$-cycle. For example, $(1\ 2)$ is a transposition. A permutation $\sigma$ is even if it can be written as a product of even number of transposition. We need to prove that given any $\sigma \in S_n$, $\sigma ^2$ is even. We have the following result.

Use the above theorem, to write $\sigma$ as a product of $m$ transpositions. \[ \sigma = \left( x_1\ \ x_2 \right) \left( x_3 \ \ x_4 \right) \dots \left( x_{k-1}\ \ x_k \right). \] So, \[ \sigma ^2 = \underbrace{\left( x_1\ \ x_2 \right) \left( x_3 \ \ x_4 \right) \dots \left( x_{k-1}\ \ x_k \right)}_{m} \underbrace{\left( x_1\ \ x_2 \right) \left( x_3 \ \ x_4 \right) \dots \left( x_{k-1}\ \ x_k \right)}_{m}. \] Thus, $\sigma ^2$ is a product of $2m$ transpositions which implies $\sigma ^2$ is even.