Would you take back a cheater if they proved to you that they've really changed?

So, some context. My boyfriend and I were together for almost 3 years. We moved to another state together, and we had been living together for the last year. There were a lot of things coming in between our relationship, like how we were living with my mom (awk, I know, things were complicated and we couldn't afford to live on our own yet), we didn't have the same days off so we hardly saw each other, etc. It got to the point where he only wanted me around for sex in exchange for him doing things for me like cooking dinner or driving me to work which was down the street. It didn't feel like we were a couple anymore.

About a month ago, my coworker was worried about my well-being and expressed how he always thought I was pretty and how he wanted the best for me, and it was nice hearing that from somebody at the time. I ended up inviting him over to the house while my boyfriend was at work and we did sexual things. My boyfriend ended up finding out what happened after looking through my phone and I was trying to explain why I felt the need to do that, but he wasn't believing me. I broke his trust. I owned up to what I did but he was too hurt and has been hurting ever since and it kills me. I've started therapy and got an STD test which came back negative, and I'm working on myself to become a better person, but I feel like it's too late since he already closed his heart off to me.

TLDR: I cheated with my boyfriend of 3 years and he doesn't trust me and doesn't want to take me back. I'm having trouble moving on because he was my everything.

What should I do? Should I keep trying to fix our relationship? How can I prove to him that I've changed and I won't ever do what I did to him again after going through the pain of losing him? How can I get him to trust me again? If you've been cheated on, did you work things out with your partner? I still feel like a shitty human being, but I'm trying to fix what I did wrong.. any help or advice would be appreciated. :(
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Maybe after some time and space
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Would you take back a cheater if they proved to you that they've really changed?
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