It's so frustrating trying to argue with some people. Some times they are too stupid to realize their argument sucks. Other times as you laid out they are personally invested in their position and will not change their mind regardless of the amount of evidence you present.
I generally try to avoid getting into lengthy arguments on the internet. Once I realize the other side is being unreasonable I usually end the conversation. It just ends up becoming a waste of time. Of course this usually results in the other person declaring victory.....