I am trying to show that $A_n$ is generated by the 3-cycles in $S_n$. It seems that every 3-cycle of the form $(a_1,a_2,a_3)$ can just be written as $(a_1,a_3)(a_1,a_2)$ so every 3-cycle turns into an even number of transpositions (2-cycles). Is this sufficient?
An element of $A_n$ is a product of an even number of transpositions. For any pair of transpositions, find one or two 3-cycles whose product is equal to their product. After doing this you can generate all products of an even number of transpositions.