Elementary Proofs of Some Stirling Bounds

We give elementary proofs of several Stirling's precise bounds. We first improve all the precise bounds from the literature and give new precise bounds. In particular, we show that for all $n\ge 8$ $$\sqrt{2\pi n}\left(\frac{n}{e}\right)ⁿ e^{{\frac{1}{12n}-\frac{1}{360n3+103n}}} \ge n!\ge \sqrt{2\pi n}\left(\frac{n}{e}\right)ⁿ e^{{\frac{1}{12n}-\frac{1}{360n3+102n}}$$} and for all $n\ge 3$ $$\sqrt{2\pi n}\left(\frac{n}{e}\right)ⁿ e^{{\frac{1}{12n+\frac{2}{5n}-\frac{1.1}{10n3}}}} \ge n!\ge \sqrt{2\pi n}\left(\frac{n}{e}\right)ⁿ e^{{\frac{1}{12n+\frac{2}{5n}-\frac{0.9}{10n3}}}.$$}