You may or may not be asked to show proof online; luck of the draw I guess.
In-store you're normally asked to show a proof of status either via a student ID or an acceptance letter from the university. Sometimes you can just be up-front about it, "I'm not a student anymore, but can I guess the student discount anyways?" If you're nice about it, you might get the discount, depends on the sales person.
Just a heads up if you apply the student discount, you can't get apply it as a business purchase.