I don't have a problem with doing the epsilon-delta proof anyway, but to actually require one seems a bit, well, fussy to me. As if advanced methods should not be relied upon, even though they rest upon a firm basis.
Consider, for example, the methods to find a derivative. I am quite happy to accept the "shortcut" formulas without referring directly to the definition of the derivative for each problem because each of the short-cut formulae have been proven. I don't demand that each new derivative I see be done the long way if I have a shorter method that has been proven. Of course, we CAN always do the problem the long way, and it's a good exercise, but surely even the meanest of professors wouldn't demand that I take the derivative of by using the definition of a derivative.