real analysis - Showing that $frac{sqrt[n]{n!}}{n}$ $rightarrow frac{1}{e}$

Show:$$\lim_{n\to\infty}\frac{\sqrt[n]{n!}}{n}= \frac{1}{e}$$

So I can expand the numerator by geometric mean. Letting $C_{n}=\left(\ln(a_{1})+...+\ln(a_{n})\right)/n$. Let the numerator be called $a_{n}$ and the denominator be $b_{n}$ Is there a way to use this statement so that I could force the original sequence into the form of $1/\left(1+\frac{1}{n}\right)^n$

Answer

I would like to use the following lemma:

If $\lim_{n\to\infty}a_n=a$ and $a_n>0$ for all $n$, then we have
$$
\lim_{n\to\infty}\sqrt[n]{a_1a_2\cdots a_n}=a \tag{1}
$$

Let $a_n=(1+\frac{1}{n})^n$, then $a_n>0$ for all $n$ and $\lim_{n\to\infty}a_n=e$. Applying ($*$) we have
$$
\begin{align}
e&=\lim_{n\to\infty}\sqrt[n]{a_1a_2\cdots a_n}\\
&=\lim_{n\to\infty}\sqrt[n]{\left(\frac{2}{1}\right)^1\left(\frac{3}{2}\right)^2\cdots\left(\frac{n+1}{n}\right)^n}\\
&=\lim_{n\to\infty}\sqrt[n]{\frac{(n+1)^n}{n!}}\\&=
\lim_{n\to\infty}\frac{n+1}{\sqrt[n]{n!}}=\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}}+\lim_{n\to\infty}\frac{1}{\sqrt[n]{n!}}=\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}}
\end{align}\tag{2}
$$

where we use (1) in the last equality to show that
$
\lim_{n\to\infty}\frac{1}{\sqrt[n]{n!}}=0.
$

It follows from (2) that
$$
\lim_{n\to\infty}\frac{\sqrt[n]{n!}}{n}=\frac{1}{e}.
$$