The integral form definition of the Gamma function is as follows. It is valid for all complex numbers with $\mathrm{Re}(z)>0$:
$$\Gamma(z)=\int_0^\infty x^{z-1}e^{-x} dx$$
See this Wikipedia page for justification.
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.
$$\Gamma(z)=\frac{1}{z}\prod_{n=1}^\infty\left[\frac{1}{1+\frac{z}{n}} \left(1+\frac{1}{n}\right)^z\right] $$
See this Wikipedia page for the justification.
How do we rigorously prove that these two definitions of $\Gamma(z)$ are equivalent on their common domain $\mathrm{Re}(z)>0$?