How to evaluate $\lim_{n\to\infty}n!/n^{k}$ I do not know how to go about finding the limit of this sequence. I know it diverges to infinity yet I can't find terms to use the squeeze lemma effectively.
$a_n = \frac{n!}{n^{1000}}$ 
$\lim_{n\rightarrow\infty} \frac{n!}{n^{1000}}$
I know that when $n>1000$, the numerator will "run out" of denominators and the expression will grow, yet I don't know how to formally prove this divergence.
In addition to this, since it is a sequence, no series laws can be applied. 
Anyone have an idea on how to approach this type of problem?

This is probably overkill, but we can use Stirling's Approximation that
$$n! \sim \sqrt{2\pi n} \left(\frac{n}{e}\right)^n$$
Then, we have
$$\lim_{n\to\infty} \sqrt{2\pi n} \frac{\left(\frac{n}{e}\right)^n}{n^k}$$
which certainly diverges to $\infty$ if 
$$\lim_{n\to\infty} \frac{n^n}{e^nn^k}$$
does. This is equal to
$$\lim_{n\to\infty} \frac{n^{n-k}}{e^n}$$
and since the denominator is of a lower asymptotic order than the numerator (we can use the ratio test to get that $n$ increasing by $1$ increases the expression inside the limit by a factor of at least $\frac{n}{e}$), we get that this diverges, and the original must as well.
