As many points as you want to whoever can answer my question thoughtfully?

So here is the background.

I dated a girl for about 4 months, by the end of it it was strictly friends with benefits because neither of us cared much. She left me because she had feelings for her ex again. That was fine, I was hanging out with lots of girls. Then we went on a weekend away together to a music festival. One of the nights we headed home early, grabbed some beer and drank and talked for hours. It came out that I didn't think there was such a thing as love, turns out that is basically what her life revolves around. She cried, I felt bad, we were friends from then on strictly as you can imagine. This was 2-3 months ago and I can't shake my feelings for her, all my other female friends have vanished cause I don't make time for them because of her. Today just casually talking she said we will never be together, kinda jokingly I said why not, and she was like cause of what happened at the festival. I care for her so much, I still am unsure about the love thing but I can't get her out of my mind and I'd love to be with her again. How can I do this at this point? How can I prove to her that I've changed?