Here's the outline of the proof I propose:
Let $\sigma \in S_n$ be any even permutation. Then by definition, $\sigma$ can be written as a product of even number of transpositions in various ways. Write $\sigma$ as a product of transpositions in one such way. Now, pair the first transposition with the second transposition, and the third with the fourth and every $(2n-1)$th transposition with the $(2n)$th transposition. Perform the following operation in every pair; $(xz)(xy)=(xyz)$ where $y\ne z$ and $(xy)(ab)=(ayz)(xab)$ where $x\ne y\ne a\ne b$. Thus, we've shown every even permutation is a product of one or more cycles of length $3$.
Is my proof correct?
Are there any alternative and neater ways to prove this?