Activity Selection - Inductive Step

Inductive hypothesis: After $k$ activities, greedy's $k$-th activity ends at time $g_k$, and OPT's $k$-th activity ends at time $o_k$, with $g_k \leq o_k$.

Inductive step: After greedy picks its $(k+1)$-th activity: Greedy picks the earliest-ending activity that starts after $g_k$. OPT picks some activity that starts after $o_k$. Since $g_k \leq o_k$, any activity OPT can pick, greedy could also consider (it starts late enough). But greedy picks the earliest-ending one. So $g_{k+1} \leq o_{k+1}$. Greedy stays ahead.