Prove that if $α ∈ S_n$ is an $m$-cycle, then $α$ is a product of transpositions.

Question

Prove that if $α ∈ S_n$ is an $m$-cycle (i.e has the form $(a_1 a_2 . . . a_m)$ for distinct integers $a_1, . . . a_m ∈ \{1, . . . n\}$ then $α$ is a product of transpositions (i.e., permutations of the form $(i j)$ for some $1 ≤ i ≠ j ≤ n$.

I can't really see how to begin this proof. I understand that this arbitrary m-cycle is a product of transpositions intuitively but I am not really sure how to prove it.

Answer 1

One can easily verify that $(a_1\ a_2 \dots a_m) = (a_1\ a_m)(a_1\ a_{m-1})\cdots(a_1\ a_3)(a_1\ a_2)$.