We can get a permutation p from the trivial permutation
(1,2,...,n) by a sequence of, say, k, transpositions.
Prove that each transposition
changes the number of inversions in the permutation by an odd number.
Deduce that if the number k is even then the number of inversions in p is
even and if k is odd then the number of inversions in p is odd.
This implies the theorem.