I have to say, I don't like how death is depicted here. She's trying to convince him that death, and by extension life, is significant. But she calls him selfish for wanting to do for feeling like his life is worthless when its clear that's what being ingrained into him by everyone around him, and that his hurt honor, especially in cultures in East Asia where success and honor reflect upon your entire family.
It raises the question of what if he was a nihilist and didn't care about living or dying? What if he throw the game from the start and stopped caring because he truly wished to die? Surely that would mean this whole charade between life and death would be pointless and she'd be wasting her time. It's funny because if she met an existentialist, she'd be fucked.
Also, how much you wanna bet that the 13th bullet takes him to right before he died?