I mean, he challenged me to prove that my mother existed, without seeing her. Obviously I couldn't.

Beliefs pay rent in anticipated experience. Your belief that your mother exists means that you would expect to see her if you went to wherever she lives or whatever. So he wants proof? Bring him to her. Can't be done in class though.

Of course he might well have meant something entirely different by "exist" than I assumed above, but the ordinary interpretation of "my mother exists" (when she's not in the speaker's presence) is something like, "I have a model in my head saying that we'll see my mother if we go to [location]".

If he means something else though, then you'll simply have to ask what that something else is.