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Symmetric Group $S_n$ is generated by transpositions.

I have to proof, that a symmetric group $S_n$, with n>=2 with support {a1, a2, ... , ap} such that σ(ak)= ak+1 (when k

I started by verifying that σ(1) • (1,2) • σ(2) = (2,3) Transposition. Then by iteration I show that the next Transposition (3,4) = σ(1) • (2,3) • σ(2)... Iterating until Transposition (n,1). after this how can I then proof that $S_n$ is generated by this Transposition?

Thanks!