It sounds like he understands he has hurt you, which is a positive happening. My instinct is that he will respect the boundaries you two have discussed.
However, I was going to make the same suggestion with blunt force. Cheaters are particularly fond of prepaid cell phones. I would actively search for any prepaid phones; favored hiding places are, of course, in pockets of clothing, in closet, in vehicle. Careful cheaters do their emailing through web based email account from smartphones or office computers so there is no trace to be found by spouse. Also I would be suspicious if he withdraws from you or suddenly starts overindulging you with gifts and flowers or asking you to do more adventurous amorous activities.