**Proof of Theorem.**

We know already: The number of permutations of {1,2,...,n} is n!

Let f be a permutation.

(a) Consider the permutations f^{0}, f^{1}, f^{2},
..., f^{n!}

(b) There exist 0 ≤ i < j ≤ n!
with f^{i} = f^{j}.

(c) Is f^{i} = f^{j} with 0 < i < j, then also
f^{i-1} = f^{j-1}.

(d) Therefore f^{0} = f^{j-i} (and 0 ≤ i < j).

Also: Since j ≤ n!, we have j-i ≤ n!