## A hard limit proof

I am clueless about how to solve the following problem:

Prove that

$\lim \limits_{n \to \infty} \frac{n!}{n^{n+(1/2)e^{-n}}}=c$, where $c$ is a constant.

Here's the hint provided:

Use the monotonicity of the logarithm to establish that
$\int_{k-1}^{k} \log{x} dx < \log{k} < \int_{k}^{k+1} \log{x} dx, k=1,...,n,$
and hence
$\int_{0}^{n} \log{x} dx < \log{n!} < \int_{1}^{n+1} \log{x} dx$