Understanding a critical inequality in Emil Artin's derivation of $\Gamma(x) = \lim_{n\rightarrow\infty}\frac{n^xn!}{x(x+1)\cdots(x+n)}$

Question

Assume that $0<x\le1$ and $n$ a positive integer greater than $2$ so that $$f(x+n) = (x+n-1)(x+n-2)\cdots (x+1)xf(x) \tag{2.5}$$ and

This final inequality is troubling me. Why are we allowed to replace $n-1$ by $n$?

Once we obtain this final inequality the derivation becomes obvious given that

$$f(x)\frac{n}{x+n} \le \frac{n^xn!}{x(x+1)\cdots(x+n)} \le f(x)$$

so that we may apply the squeeze theorem.

Answer 1

You can think more simply as follows: If $g(n) < b$ for all $n$, then $g(n+1)< b $ for all $n$ as well. The reason why you can replace is that $f(x)$ in the above bounds does not depend on $n$.