Why is he being so stubborn about leaving her?

A nice, quiet man is in a relationship with this woman who has two kids. It all started with her car breaking down and he decided to help her. Then, he ended up having feelings for her and is very attached to her kids.She treats him like crap. She won't let him hug, kiss, or hold her hand. They argue everytime he asks her a question and she is very short tempered. She won't have sex with him either. She makes excuses...headache or acts like her cellphone is ringing. He helps her out with the bills and kids. He's been with her for nine months now. He thinks she's cheating, but won't leave til he catches her. He has said to her : I believe you are cheating on me and she said, usually the one saying it is the one doing it. I guess catching her is his way of getting her back. Before he got with this woman, he had a gf for 6 years. She was the best one ever. He wants her back, but will not leave this woman until he catches her cheating. Will he ever give up? Why does he want to catch her so bad?