Time and consistency, like everyone is telling you. Don’t beat yourself up about it, though. You can’t let his lack of trust make you doubt yourself. When he says these things, I don’t think it’s necessary to get defensive. You can understand it as an expression of his fears, and take it as an opportunity to comfort him, without saying anything. It doesn’t require a response. It’s just his thinking, and you can’t do a thing about it, except to help him learn to detach himself from the thought.
You let him have it. You can even validate it (“yeah, it’s horrible when people let you down”). But you don’t have to let it throw you. He’s talking about someone else. Not you. If you understand that, it should let you stop worrying so much about proving yourself. You don’t need to prove yourself (you know yourself), and further, you can’t do it. Only he can do it in his own head.
Time will tell. If he keeps on distrusting you forever, it means he’s trying to drive you away. This could be due to a thinking disorder of some kind, or it could be a lack of self-esteem, or even self-hatred. He may not believe he deserves someone who is faithful to him.
Anyway, like I say. You know yourself. Be yourself, and let him worry about his trust issues. If you fall into trying to deal with them, you’ll be trapped in an endless loop, and eventually it will become so hurtful, that you’ll prove he’s right by leaving him.