I do not find the following statement obvious. The context is to show the gauss product representation of gamma function satisfying one of the characteristic properties of gamma function being bounded in $1\leq x<2$.
For large $n\in N,1\leq x<2$, $\frac{n^{-x}}{n!}x(x+1)\cdots(x+n)>a>0$ for some $a\geq 0$. This bound should be universal bound for large $n$.
I tried by $\frac{x}{1}\frac{x+1}{2}\cdots\frac{x+n-1}{n}\frac{n}{n^{x}}$. However $1\leq x<2$ does not grant me permission to conclude that this is non-zero.
Is this fact obvious?