Does this sequence converge or diverge

a(subscript)n = n!/n^n (hint: compare with 1/n)

Note that for all integers n > 0
n!/n^n = (1 * 2 * 3 * ... * n) / (n * n * n * ... * n)
...........≤ (1 * n * n * ... * n) / (n * n * n * ... * n)
...........= n^(n-1) / n^n
...........= 1/n.

Hence, 0 ≤ n!/n^n ≤ 1/n.
Since lim(n→∞) 1/n = 0, we conclude by the Squeeze Theorem that lim(n→∞) n!/n^n = 0.

I hope this helps!