Showing $n! = \sqrt{2\pi n} (\frac{n}{e})^n \big(1 + \mathcal{O}(\frac{1}{n})\big)$ from $\log(n!) =n\log(n) - n + \mathcal{O}\big(\log(n)\big)$

Question

I wish to prove Stirling's Formula in this way, in particular showing the first term in the series is $\mathcal{O}\big(\frac{1}{n}\big)$, and I've come across a "proof" that simply states there is "more work needed" between these two statements, and I can't work out how to do it. Would really appreciate any help, thanks.