Dealing with bf (maybe ex) who accuses

For a lot of my relationship my bf has had horrible trust issues. He would accuse me of lying and cheating when that is no where near anything that even crosses my mind. He would need screenshots of call logs and to Skype to see me sometimes to make him understand that I'm not doing anything crazy. Sometimes it wouldn't be enough or he would apologize and recognize that he was over the top. It's been almost a year and it hurts that he doesn't trust me and can't put faith in that I'm honest and have no intention of hurting him

Here's my question. We have pretty much been breaking up since yesterday because he really thinks I have lied to him and am doing something behind his back. I have no evidence for him to show him otherwise. This time is different and really feels like the end. I have told him my truth. How do I just let it go and him go and let him just think that about me forever??