Teach doesn't understand that kind of person, he would only come back again and again if you let him live. Like a batman and joker situation, joker has killed thousands of people just because batman refuses to get rid of him.
@darkdrake You could see it one of 2 ways, Hajime being nice and taking the blame for his death to keep his teacher from blaming herself (which is what Yue says to him) or, as him killing an enemy since that guy was never gonna be an ally. levyoasu could probably give a better explanation, since I haven't read the LN myself.
Killing him is what's merciful here, if he had wanted to be mean, he'd have patched him up and made him a slave, working him to the brink of madness. He'd have been useful that way too, and the teacher would have been happier. I really don't get some authors, they try to portrait a vengeful character but end up being dumb saints as well as wasteful people.