In the proof that i am reading, the "strategy", that is used to prove stirlings approximation, is to show that $$\lim_{n\to\infty}\frac{n!}{\sqrt{2\pi n} (\frac{n}{e})^n }=1.$$
But a quick check shows that the error between factorial and the approximation does not monotonically decrease. So how can we use this limit to show that it's an accurate approximation. In fact, how do we even "proof", an approximation, as we haven't defined "accurate". Why would $\sqrt{2\pi n} (\frac{n}{e})^n$, be a more accurate approximation of $n!$, than for example $(1.0001n)!$
This is an equivalent in the sense of asymptotic analysis. Here, accurate simply means that the difference between $n!$ and $\sqrt{2\pi n}n^n e^{-n}$ is much smaller than $n!$ when $n$ tends to infinity. In fact this difference divided by $n!$ tends to zero.
The other aspect of the approximation is that the expression $\sqrt{2\pi n}n^n e^{-n}$ belongs to an asymptotic scale of functions that involve only polynomials, logarithms and exponential. In this respect, it can be considered a simpler expression than $n!$ and it can be used to prove other results.