This is a really nice problem.

Now, I agree an Opalg's argument.

3/4 still remains in the OP's cantor set. As n goest to infinity, the radius of an open interval at 3/4 goes to 0, but 3/4 is intact.

As mentioned by Opalg, the below is a key idea that OP's cantor set is not empty,

"It may appear thatonlythe endpoints are left, but that is not the case either. The number 1/4, for example is in the bottom third, so it is not removed at the first step, and is in the top third of the bottom third, and is in the bottom third ofthat, and in thetopthird ofthat, and so on ad infinitum—alternating between top and bottom thirds. Since it is never in one of the middle thirds, it is never removed, and yet it is also not one of the endpoints of any middle third. (wiki) "