How do you prove to someone that you love them? (when they really don't really trust anyone as a result of having been crushed by an ex-boyfriend)

yes .. This is a complicated situation I see you are in , you know people have memories , emotional scars ,physical scars  and there are many people that take some time to open up to you , they have usually had something bad sad happen to them in order to make them not trust anyone , maybe a failed past relationship , betrayal etc. It must be frustrating sometimes when they don't seem to trust you at all , or they seem to hiding things from you , but really your love for them means sticking it with her , not doing something bad that will make her not trust you even more , be patient and loving and caring , and show her you care for her and that you aren't going to hurt her , don't ask her already what has happened in that relationship , because really who wants to revive some bad horrible experience they have had , so even though It may take a lot of time , to get her to slowly come out and tell you things that she can't talk about be caring be accepting don't blame her for it , comfort her be there for her , because that will really show that you love her .