Using the results of the previous questions (in particular the integral representation of $I_n$ from question 2, the bounds from question 3, and the Gaussian integral $\int_{-\infty}^{+\infty} e^{-x^{2}/2}\, dx = \sqrt{2\pi}$), deduce Stirling's formula:
$$n! \underset{n \rightarrow \infty}{\sim} \sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}.$$