Basis : If n= 4, n! = 24 >= n^2 = 16.
Inductive step : Let's suppose n! >= n^2 is true for all n>=4, n in NN. Let's show that it also holds for n+1 > 4.
(n + 1)! = n! * (n+1) >= n^2 * (n + 1), by induction hypothesis
>= 4*n^2, because n+1 > 4 by hypothesis
>= n^2(1 + 2/n + 1/n^2), because n^2 >= n > 1 by hypothesis
= n^2 +2n + 1 = (n + 1)^2.
You can now conclude by mathematical induction that n! >= n^2 AA n>= 4, n in NN iff 0 <= 1/(n!) <= 1/n^2 AA n>=4, n in NN.
Since the series of general term 1/n^2 converges, you can conclude by the comparison test that the series of general term 1/(n!) also converges.
