Can he change?

My bf and I were on and off for a few months when we began dating. He claimed to just be friends with his ex which after looking through his fb on his phone that he constantly hides, I found he was begging his ex to come back to him. They were at the very least spooning and flirty. The messages ended a month before we moved in together (found out a month  after moving in). I demanded he never  have any contact with her again, which I don't think he has (it's now 4 months later). His explanation at the time was that we were so on and off, he didn't think we would last. He loves me. I love him. We spend every waking moment together and are mostly happy. But this always plays on my mind as how we started. Don't know that I can ever truely trust that it's in the past. Should I try and forget about this? Will he be loyal?