Alternating Group is Generated by 3-Cycles

Theorem
Let $n \in \N$ such that $n \ge 3$.

Let $A_n$ denote the alternating group on $n$ letters.

Then $A_n$ can be generated by the set of $3$-cycles:
 * $\set {\tuple {1, i, n}, i \in \set {2, 3, \ldots, n - 1} }$