Euler Form of Gamma Function at Positive Integers

Theorem
The Euler form of the Gamma function:
 * $\displaystyle \Gamma \left({z}\right) := \lim_{m \mathop \to \infty} \frac {m^z m!} {z \left({z + 1}\right) \left({z + 2}\right) \cdots \left({z + m}\right)}$

converges to the factorial function at positive integers:


 * $\displaystyle \lim_{m \mathop \to \infty} \frac {m^n m!} {\left({n + 1}\right) \left({n + 2}\right) \cdots \left({n + m}\right)} = n!$

Proof
Now we have from Factorial of Integer plus Reciprocal of Integer that:


 * $\displaystyle \lim_{m \mathop \to \infty} \dfrac {\left({m + n}\right)!} {m! m^n} = 1$

Now: