# Gamma Difference Equation/Proof 2

$\map \Gamma {z + 1} = z \, \map \Gamma z$
 $\displaystyle \frac {\map \Gamma {z + 1} } {\map \Gamma z}$ $=$ $\displaystyle \paren {\frac 1 {z + 1} \lim_{m \mathop \to \infty} \prod_{n \mathop = 1}^m \frac {\paren {1 + \frac 1 n}^{z + 1} } {1 + \frac {z + 1} n} } \div \paren {\frac 1 z \lim_{m \mathop \to \infty} \prod_{n \mathop = 1}^m \frac {\paren {1 + \frac 1 n}^z} {1 + \frac z n} }$ $\displaystyle$ $=$ $\displaystyle \frac z {z + 1} \lim_{m \mathop \to \infty} \prod_{n \mathop = 1}^m \paren {\frac {\paren {1 + \frac 1 n} \paren {z + n} } {z + n + 1} }$ after some more-or-less hairy algebra $\displaystyle$ $=$ $\displaystyle z \lim_{m \mathop \to \infty} \frac {m + 1} {z + m + 1} = z$
$\blacksquare$