Alternating group as a product of 3-cycles

Show that An for n≥3 is generated by 3-cycles, i.e., any element can be written as a product of 3-cycles.

My solution is quite simple, but I wonder if I've missed something:

Any element of an alternating group can be written as a product of even number of 2-cycles. Let's consider pairs of 2-cycles in such expression. They can be of two forms:

(a,b)(b,c) and (a,b)(c,d).

The first is immediately a 3-cycle: (a,b)(b,c)=(a,b,c).

The second can be made into 3-cycles like this: (a,b)(c,d)=(a,b)(b,c)(b,c)(c,d)=(a,b,c)(b,c,d).

So, each pair of 2-cycles can be converted to one to two 3-cycles.

Edited January 28Jan 28 by Genady