# Proving that the Gamma function infinite product definition extends its integral form definition

The integral form definition of the Gamma function is as follows. It is valid for all complex numbers with $\mathrm{Re}(z)>0$:

$$\Gamma(z)=\int_0^\infty x^{z-1}e^{-x} dx$$

It is well-known that the Gamma function also has this infinite product expression that is valid for all complex numbers $z$ except for the negative integers.
$$\Gamma(z)=\frac{1}{z}\prod_{n=1}^\infty\left[\frac{1}{1+\frac{z}{n}} \left(1+\frac{1}{n}\right)^z\right]$$
How do we rigorously prove that these two definitions of $\Gamma(z)$ are equivalent on their common domain $\mathrm{Re}(z)>0$?