Inductive hypothesis: After activities, greedy's -th activity ends at time , and OPT's -th activity ends at time , with .
Inductive step: After greedy picks its -th activity: Greedy picks the earliest-ending activity that starts after . OPT picks some activity that starts after . Since , any activity OPT can pick, greedy could also consider (it starts late enough). But greedy picks the earliest-ending one. So . Greedy stays ahead.