Should a cheater be forgiven?

I found an email from my boyfriend talking to his ex crush, and in it he said "in the end u avoided me" and it probably doesnt matter, but i still care about u alot, well i emailed this person and she told me she was not interested in him and had no intention of getting with him because she was in a relationship. Also before this instance my boyfriend emailed someone else and let that person borrow 150 dollars and said he only let her borrow it because he said she told him that someone had snatched her purse or watever do u guys think it was more money that he probably let her have than what he really told me. Also i am really finding it hard to move past this even though he helps me financially, i know that some guys will do this in the attempt to keep a girl so i need to know how to dump this person, ive already told this person that i want to break up but he insists that he will make it up, and claims that he loves me tenfold, and that he is no longer doing that behind my back. I asked him why he lied to me and his response was because i didnt want to hurt you. So how do u unmask this persons true intentions? He says he isnt cheating because hes always home with me. But how do i know if hes not talking to anyone besides me perhaps even at work? I dont trust this person at all. How do i tell him i cant forgive him?