Equivalence of Definitions of Gamma Function

Theorem
The following definitions of the Gamma function are equivalent:

Weierstrass Form equivalent to Euler Form
First it is shown that the Weierstrass form is equivalent to the Euler form.

Combining the limits:

But:
 * $(1): \quad m = \dfrac {m!} {\left({m - 1}\right)!} = \dfrac 2 1 \cdot \dfrac 3 2 \cdots \dfrac {x+1} x \cdots \dfrac m {m - 1}$

Each term in $(1)$ is just $\dfrac{x+1} x = 1 + \dfrac 1 x$, so:
 * $\displaystyle m = \prod_{n \mathop = 1}^{m-1} \left({1 + \frac 1 n}\right)$

Thus the expression for $\dfrac 1 {\Gamma \left({z}\right)}$ becomes:

Hence:
 * $\displaystyle \Gamma \left({z}\right) = \frac 1 z \prod_{n \mathop = 1}^{\infty} \left({1 + \frac 1 n}\right)^z \left({1 + \frac z n}\right)^{-1}$

which is the Euler form of the Gamma function.

Integral Form equivalent to Euler Form
This is proved in the page:
 * Integral Form of Gamma Function equivalent to Euler Form