Limit of $\frac{n!}{2^{(n^{2})}}$ 
Find the following limit:$$\lim_{n\to\infty} \frac{n!}{2^{n^2}}$$

I get the feeling that this limit equals to zero. Intuitively, the function $f(n)=2^{n^2}$ grows much more faster than the factorial, however, I wish to prove this limit using only the squeeze theorem or some algebra. I noticed that: 
$$0\leq \frac{n!}{2^{n^2}}=\frac{n}{2^n} \cdot\frac{n-1}{2^n}\cdots \frac{2}{2^n}\cdot\frac{1}{2^n}$$
I tried to think about cases about wether $n$ is even or odd, hoping that would lead me to a way to simplify the latter expression, but it didn't work. Also, is there a way to generalize the problem? That is, does the limit
$$\lim_{n\to\infty} \frac{n!}{a^{n^a}}$$ is always equal to zero, for $a\in\Bbb{N}$? 
Thanks in advance!
 A: $$n! \leq n^n=2^{n\log_2 n} $$
This should give you your squeeze theorem argument. 
A: It is easy to see that  $$n/2^n \leq 1/2$$ for each integer $n \geq 1$. 
So we can limit $a_n = \frac{n!}{2^{n^2}}$ (from above) with a geometric progression, I think.    
How? Prove the above inequality by induction.   
Then using that inequality we have:   
$$\frac{n!}{2^{n^2}} \leq \frac{n^n}{2^{n^2}} = (\frac{n}{2^n})^n \leq (1/2)^n$$
Then we can find that the limit is zero using the squeeze theorem.    
A: By ratio test
$$\frac{\frac{(n+1)!}{2^{(n+1)^2}}}{\frac{n!}{2^{n^2}}}=\frac{n+1}{2^{2n+1}}\to 0$$
and the latter can be easily proved by induction or ratio test again.
Edit: Note that ratio test is essentially equivalent to the squeeze theorem ( see the proof).
A: We can prove directly by induction proving that $n!\le2^{n^2-2n+1}$, indeed


*

*base case: $n=1 \implies 1\le 1$

*induction step: we need to prove
$$k!\le2^{k^2-2k+1}\implies (k+1)!\le2^{k^2}$$
which is true indeed
$$(k+1)!=(k+1)k!\le(k+1)2^{k^2-2k+1}\le2^{k^2-2k+2}\le 2^{k^2}$$
then
$$n!\le2^{n^2-2n+1}< 2^{n^2}$$
and then
$$\frac{n!}{2^{n^2}}\le \frac1{2^{2n-1}}< 1$$
A: Seemingly a bit late this answer but I think it is worth mentioning it.
A possible way is as follows:


*

*Consider first 
$$a_n = \sqrt[n]{\frac{n!}{2^{n^2}}}=\frac{\sqrt[n]{n!}}{2^n}\stackrel{GM-AM}{\leq} \frac{\frac{n(n+1)}{2n}}{2^n} = \frac{n+1}{2^{n+1}}\stackrel{2^{n+1}=(1+1)^{n+1}}{\leq}\frac{n+1}{\binom{n+1}{2}}=\frac{2}{n}\stackrel{n\to \infty}{\longrightarrow}0$$
Since $\lim_{n\to \infty}a_n = 0 \Rightarrow \lim_{n\to \infty}a_n^n = 0$.
Your general case for $a\in\mathbb{N}, a \geq 2$ follows immediately from the case $a=2$, since $\frac{n!}{a^{n^a}}\leq \frac{n!}{2^{n^a}}\leq \frac{n!}{2^{n^2}}$.
