Cheated on my ex but he wants me to try to win him back

I 21F cheated on my ex 20M by telling someone online I liked them but it never went on after that, never engaged in a relationship it was mostly because the guy kept on sending me money lol I wasn't emotionally involved but still that's cheating.

So, we broke up and he is already kind of talking to people right now. He said it's even working out with some of them. But he also said that he still loves me and cares for me, he wants me to try and prove myself to him. I mean I could, I loved him so much and I deleted all my other socials except the one that I talk to him on. I've disciplined myself to better myself as a person overall.

Should I keep on trying (it is kinda of hard because I text and try to talk to him a lot but he rarely talks to me, he leaves me on seen most of the time but if I stop he tells me to keep on trying if I want to win him back) or should I just let this go and be a better person for the next guy that I will date in the future?