We recall that the sequence $\left(\left(\sum_{k=1}^{n} k^{-1}\right) - \ln(n)\right)_{n \geqslant 1}$ converges. We denote $\gamma$ its limit. Let $n \in \mathbb{N}^*$. For $x \in ]0, +\infty[$, we set
$$\Gamma_n(x) = \frac{1}{x} e^{-\gamma x} \prod_{k=1}^{n} \frac{e^{x k^{-1}}}{1 + x k^{-1}}.$$
Show that the sequence of functions $\left(\Gamma_n\right)_{n \geqslant 1}$ converges pointwise on $]0, +\infty[$ to a function $\Gamma$ from $]0, +\infty[$ to $]0, +\infty[$.