I'm reading the following proof from Lang's book of complex analysis. But I don't know why is it necessary that $\alpha<1$?. I don't know where "$\alpha<1$" is used . I don't understand either why $|g_{\epsilon}(s)|\to 0$ as $|s| \to \infty$. Can you help me, please?. I have been stuck for many hours.
Thank you for any help.
