My boyfriend cheated..am I doing the right thing by taking him back?

My boyfriend and I have been together for about 5 months now. I have known him for about 10 years though. Let me start off by saying that he has a very big drinking problem and I did not realize that until about a month into our relationship. Everything was going great in the beginning I was very attracted to him and very much in love with him and still am. During the summer I started realizing that when the weekend would roll around he would never make plans to be with me. He would always go to his shore house or out with his friends so he would say. I trusted him because I had no reason not to. His drinking would get really out of control he would drink every single night without fail and would get extremely drunk to the point where he would turn into a monster and we would always get into fights and break up. Then the next morning he would apologize and tell me how much he loves me and I would forgive him but then one week later it would happen all over again. He told me every single day that he wants to marry me and that I am the best thing to ever happen to him. long story short, I caught him cheating on me last month. he told me he was going to a friends and I had a feeling he was not telling the truth and I drove by his house and sure enough he was in bed with another girl (I heard them outside his window) I was absolutely heartbroken and I broke up with him. The next day he called texted did everything he could to get in touch with me and I told him I was never going to take him back after what he did to me. He threatened to kill himself and check himself into rehab for me and that he would not be able to live without me in his life. He would cry hysterically to the point where his friend would have to call me because he couldn't talk. He promised me that he would STOP the drinking and change his number and delete every girl in his phone. I love the man to death and I told him if he does that and proves that he will change then I will take him back. so far its been a month and he has changed his number, not talked to any girls and has not had one sip of alcohol. we have been together every day and every night since and I can see that he his changing.

I recently saw some of his emails that he had left up on my computer by accident and I saw that during the summer while we were together he was messaging other girls on facebook (including his ex) and he was telling her that he wanted to try and 'settle' things with him but she refused. now I know that was all in the past and that he has turned over a new leaf but it is still making me sick to think that he was doing all this for the first 4 months of our relationship. I have my doubts but I also think about the huge changes in his life he has made for me. do you think I am making a mistake by taking him back? do you think he will do it again? I love him so much and I pray to god that he will never go back to his old ways but I would just really like someone else's opinion. thank you
My boyfriend cheated..am I doing the right thing by taking him back?
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