Well I held this belief after witnessing many people go back to eating meat, including people I knew well.
I believe it comes down to selflessness, becoming vegetarian for the sake of your own health (considering the fact that meat contains antibiotic trances, excessive salt, hormones etc. )is a benign act, but still inherently selfish when you come to think of it, such people may turn to "organic meat" or something similar in the future when a healthy solution is available; on the other hand, doing it for the innocent lives of animals and environment protection is much more grand cause, for the greater good if you will.
I consider vegetarianism/veganism as a lifestyle & world view, therefore I feel that merely adapting it as a diet is not enough, if views and ideals in a person don't change then the mentality isn't too much different from that of ordinary people, the constant pressure to fit into society will bring down many people back to their old habits. Good news is that Vegetarianism is being promoted more in recently years due to climate change, giving people another great reason to become and remain a Vegetarian.
With that being said, when I try to convince someone to go vegetarian I usually don't bring up animals as it'll often imply that the person is not benevolent, instead I will actually focus on the health benefits ( and the environmental effects ), they'll soon figure out the animal part themselves if they come on board.