weaker version of Stirling’s approximation

One can prove a weaker version of Stirling’s approximation without appealing to the gamma function. Consider the graph of $\ln x$ and note that

$$\ln (n-1)!\le {\int }_1^n\ln x\mathrm{d}x\le \ln n!$$

But $\int \ln x\mathrm{d}x=x\ln x-x$, so

$$\ln (n-1)!\le n\ln n-n+1\le \ln n!$$

and thus

$$n\ln n-n+1+\ln n\ge \ln (n-1)!+\ln n=\ln n!\ge n\ln n-n+1$$

so

$$\ln n-1+\frac{1}{n}+\frac{\ln n}{n}\ge \frac{1}{n}\ln n!\ge \ln n-1+\frac{1}{n}$$

As $n$ gets large, the expressions on either end approach $\ln n-1$, so we have

$$\frac{1}{n}\ln n!\approx \ln n-1$$

Multiplying through by $n$ and exponentiating, we get

$$n!\approx n^n{e}^{-n}$$