You clearly do not trust your boyfriend at all. Any response in this thread that says "he's not lying and the screenshot proves it", you have immediately questioned, asked whether it could've been faked, whether the commenter was really sure, et cetera; while the few responses that claim the screenshot is not indicative of whether he's lying or not (which, incidentally, does not automatically mean he's lying) have been immediately gobbled up by you with no second thought.

You may claim you trust him, but your actions speak far louder than your words. At this point, regardless of whether he cheated or not, it seems to me like this relationship is doomed.