Alcohol problem or just having fun?

I think my boyfriend is becoming an alcoholic. Recently he has urged me to go for a drink with him after lunch or during lunch breaks and tends to get to a point where he starts puking or throwing up when we go out with friends. At first, I thought he was just having fun, but it's happening nearly every time we go out.

His close friend takes him outside and tries to hide it but I think it's very immature and embarrassing not knowing your limit and not knowing when you should stop drinking.

I've tried taking to him about it but he just says he knows and his sorry, but I don't think he gives a toss about how I feel about the situation. Do I try and persevere with him or just break things off now before they get worse?