Sharp inequalities related with Burnside's formula

We prove the following double inequality related with Burnside’s formula for $n!$ \begin{equation*} \sqrt{2\pi}\left(\frac{n+a_}{e}\right)^{n+a_}<n!<\sqrt{2\pi}\left(\frac{n+a^}{e}\right)^{n+a^},(n\in$\mathbb{N}$), \end{equation*} where the constants $a_{\ast }=\mathrm{0.428844044...}$ and $a^{\ast }=0.5$ are the best possible. We believe that the method we used in the proof gives insight to undergraduate students to understand how simple inequalities can be established.