It's a problem I found unconventional :
Problem :
Let $0<x$ be a real the define :
$$f\left(x\right)=\prod_{n=1}^{292}\left(\frac{n+1}{x!+n}\right)^{\frac{1}{n}}$$
Shows that :
$$f(x)<1+C$$
Where $C$ is the Champernowne constant equal to $0.12345678910111213141516171819\cdots$
Taking derivative or logarithmic derivative the problem is to show that the value of the minimum of the Gamma's function is less than the well-know constant plus one .
We don't need a great accuracy and the approximation $x_{min}=0.4616\simeq\frac{\left(\pi-e+\frac{1}{2}\right)}{2}=I$ seems to works as well .
We have can also use the Euler Reflection formula
Can we pretend to show it by hand ?
Thanks in advance .