<human>: Can you summarize the given text in a few sentences?

First post on reddit, I'll try to keep it short.

Bit of background - although we get on very well, me and this guy have always been rocky and in October we broke up for a month after I couldn't stand the fact he made me upset all the time. Since getting back together in November we have been working to improve things and it has been going well.

Recently he left his Facebook logged in on my laptop, and I discovered that he has been messaging his ex-girlfriend very suggestively, reciprocating her requests for sex and implying that they will have sex soon as well as suggesting to meet up.

This doesn't come as a surprise to me - he has lied about how many partners he has had in the past, he lied to me about the girl he slept with whilst we were on our break and AFTER getting back together in November, he slept with the girl from the break once again (effectively cheating on me) and wouldn't admit it until I asked him. I don't want to end it with him because I know I will be even more miserable without him and I want to fight for this because I like him a lot, but if he does sleep with her then I know I will have to end it. 

My main issue is - how can I get him to admit that he has slept with her without admitting myself that I was on his Facebook? If I admitted that I had been monitoring his Facebook, it would also put me in the wrong and I would like to have the higher ground in this situation. Even if I tried dropping subtle hints about exclusivity and cheating, I know he would never admit it as he is a serial liar.
<bot>: boyfriend might cheat/be cheating on me, want him to admit to it without him finding out i was stalking his Facebook.