Let $n \in 2, 3, 4, \ldots$ be a positive integer. Using the results
we have the inequalities
\begin{equation}
n \log n - n
< \log n!
< n \log n
\end{equation}
Dividing each side by the nonzero quantity $n \log n$, we obtain
\begin{equation}
1 - \frac{1}{\log n}
< \frac{\log n!}{n \log n}
< 1
\end{equation}
Since $\log n \nearrow \infty$ as $n \to \infty$, then $1 - \frac{1}{\log n} \nearrow 1$ as $n \to \infty$. Thus, by the squeeze principle, we have
\begin{equation}
\lim_{n \to \infty} \frac{\log n!}{n \log n}
= 1
\end{equation}
which is what was desired to be shown. $\square$
