Howdy all, having issues with some group theory problems. Hints or suggestions are very welcomed.

(1) Prove that for n>=3 the subgroup generated by the three cycles is An.

(2) Prove that the smallest subgroup of Sn containing (1, 2) and (1, 2, . . . , n) is Sn

This is what I have so far:

Sn is generated by 2-cycles. Any element in An is a product of an even number of transpositions, so we only need to show the product of any two transpositions can be written in terms of 3-cycles.

(12)(13) = (123), while

(12)(34) = (123)(143).

Is this sufficient enough, I feel as if this is not proofy. HELP!!