How do you prove to someone that you really like them and that they should give you a chance?

Well, I was once in that situation, and take it from me, it sucks. They're is no definite way. You see, I liked this guy and we were goining out but we didn't go to the same school so we didn't get to see each other enough. One day we broke up and I was devestated. But then he switched schools and he said that I had hurt him too much so he turned me down. I gave him a picture that I drew when we first started goining out and told him how I felt (don't be too dramatic or clingy or creepy). He said that he didn't like me anymore and I asked him for a chance and he told me how he felt but he couldn't give me another chance.

So all you can do sweety is express your feelings, get close to her/him and be good friends then tell her, if she says no than ask for one chance and tell her she won't regret it and if they still say no then ask to be friends because that's the best you can do. If she doesn't like you back then she's the one who's missing out. Move on and find someone better. :)