Let $\sigma \in S_n$ be a cycle. Prove that $\sigma$ can be written as the product of at most $n-1$ transpositions.

Question

I know that $$(a_1,a_2,...,a_n)=(a_1a_n)(a_1a_{n-1})...(a_1a_3)(a_1a_2)$$

But how to prove this decomposition has the maximum number of non-repeating transpositions?

One start point may be that this decomposition contains all the symbols in the cycle, but then I have to prove all other decompositions that contain all the symbols in the cycle has no more transpositions than this one does, and I got stuck again.

I googled the question but to no avail.

Any help will be appreciated!