How can I prove that
$\displaystyle \Gamma(z)=\lim_{n \to \infty} \displaystyle \int_0^n \left( 1-\frac{t}{n}\right)^n t^{z-1}\ \text{d} t\;=\displaystyle \int_0^{\infty} e^{-t} t^{z-1}\ \text{d} t\;$
Issue is how can I prove that the order of the limit and the integral can be changed.
I know about the dominated convergence theorem and the monotone convergence theorem, but the additional problem here is that the integration limit itself depends on n.