If someone is mean to you, is it better to argue or just accept his/her point?

I think the only thing you can do with people like this is ignore them and hope they go away. Someone who is this adamant at believing they are right, will be impossible to convince. All you can hope for is that someday they will see the light and figure out they were wrong. That may not ever happen, but I believe in karma. What comes around goes around!