\begin{align*} \Gamma(n+1) & = \int_0^{\infty} u^n e^{-u} du\\ & = -u^n \cdot e^{-u} \big |_0^{\infty} + n \int_0^{\infty} u^{n-1} e^{-u} du \\ & = 0 - 0 + n \Gamma(n) \end{align*}