After the first exchange, OPT' agrees with G on the first activity. Now apply the same argument to the remaining activities. By induction, you can transform any OPT into G through a sequence of exchanges, each maintaining optimality. Therefore, G is optimal.
Here's the exchange argument in action. You found a way to swap OPT's choices for G's choices without losing quality. Since you can reach G from any OPT, G must be optimal. The proof shows that finishing early never hurts.