To show that a cycle can be written as a product of one or more transpositions, it suffices to prove the validity of the identity , which will be done by using mathematical induction.

Assume that the identity holds for . It will be shown that it also holds for , as in .

Start by noticing that .

Due to the induction assumption, which says that a -length cycle can be decomposed into a product of transpositions, can be written as .

By combining the last two equations, the conclusion follows for .