Variation on the Cantor Set

We're going to create a set that is a variation on the Cantor Set. Start with the open interval . Remove the segment . Now you have . Next remove and and you're left with . Repeat ad infinitum...

I suspect this set is empty. Proof:

Want to Prove: There are no interior points.

Assume that such that .

For to create a neighborhood contained in .

For .

For .

For .

Thus, as , we have that , which is clearly a contradiction. Thus there are no interior points in this set. However, is an open set (I'm worried this is false), so all points are interior points, leading us to the conclusion that is empty.

Is this correct?