How to prove that this limit diverges?

It holds that \begin{align*} \log n! & = \log n + \sum\limits_{k = 1}^{n - 1} {\log k} \le \log n + \sum\limits_{k = 1}^{n - 1} {\int_k^{k + 1} {\log tdt} } \\ & = \log n + \int_1^n {\log tdt} = \log n +n\log n - n, \end{align*} i.e., $$ n! \le n\left( {\frac{n}{e}} \right)^n. $$ Thus, $$ \frac{{2^{\sqrt n } }}{n} \le \frac{{2^{\sqrt n } }}{{n!}}\left( {\frac{n}{e}} \right)^n . $$ Show that the left-hand side tends to infinity as $n\to +\infty$.