Is it love? Should i forgive a cheater?

I found messages of my ex with a female trying to see eachother & him inviting her to his place to basically have a one night stand, she told him to go to her place instead.. while he was lying to me about what he was planning to do. I found out and broke it off with him 2days ago, since then he has not stopped trying to get me back says all the things possible does the most after i blocked him of everything, calls me from like 4diff numbers comes to look for me & waits for me to hive him the opportunity to see him & i do not give him the opportunity to talk to me, Because the day i broke up with him he just walked out the car, left me with words in my mouth. I didn’t get to tell him the real reason in person, just that I wasn’t happy so he walked away less than 3min into it. After i told and sent him screenshots he’s been saying all the right things possible has tried calling me every hour of the day & says hes doing bad can’t live his life without me, yet he was trying to do that to me & just come to me like nothing? Hes doing me bad by trying to get me back, idk if its because he does really love me and people “make mistakes” but I don’t see cheating as a mistake, thought goes into it, i have zero tolerance i know he in no way would have forgiven me. I do not know what to do😪

I told him that he not only put me emotionally at risk but also my health and that is no joke. That he should be more careful in the future because he doesn’t think with his head or his heart just with his pe- is.