Stays ahead works well when:
1. There is a natural measure of progress (end time, count, position..)
2. Greedy's progress is easy to compare with any alternative's progress
3. Being ahead at step k helps you stay ahead at step k+1
4. The inductive step is simple For activity selection, end time is a perfect measure. Finishing earlier gives more room for future activities. The induction flows naturally. If the comparison between greedy and others is direct, stays ahead is your tool.