Euler Form of Gamma Function at Positive Integers

Theorem
The Euler form of the Gamma function:
 * $\ds \map \Gamma z := \lim_{m \mathop \to \infty} \frac {m^z m!} {z \paren {z + 1} \paren {z + 2} \cdots \paren {z + m} }$

converges to the factorial function at positive integers:


 * $\ds \lim_{m \mathop \to \infty} \frac {m^n m!} {\paren {n + 1} \paren {n + 2} \cdots \paren {n + m} } = n!$

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


 * $\ds \lim_{m \mathop \to \infty} \dfrac {\paren {m + n}!} {m! m^n} = 1$

Now: