Proving continuous function on z

[william_33]

For each fixed \(\displaystyle \,n\,\), show that \(\displaystyle \displaystyle{\,f_n(z)\,=\,\int_1^{n}\,t^{z-1}\,e^{-t}\,dt\,}\) is a continuous function of \(\displaystyle \,z\,\) and show that \(\displaystyle \displaystyle{\,g(z)\,=\,\int_0^1\,t^{z-1}\,e^{-t}\,dt\,}\) is analytic for \(\displaystyle \,\Re{z}\,>\,0.\)

What I know is that at \(\displaystyle \displaystyle{\,g_n(z)\,=\,\int_{\frac{1}{n}}^1\,t^{z-1}\,e^{-t}\,dt\,}\) I must show that for each \(\displaystyle \,g_n(z),\,n\,\) is analytic and that \(\displaystyle \,g_n\,\) converges uniformly to \(\displaystyle \,g\,\) on closed bounded sets of \(\displaystyle \,\Re z\,>\,0\,\), but I do not know how to prove this?