Reply to Hartlw:

Why even talk about boundary points in this thread? The most direct, constructive, intuitive proof of the problem in the OP has nothing to do with boundary points: it never needs to mention boundary points.

This is manifestly false. Let

and let

Clearly,

is both an accumulation point of

and an interior point of

If you meant to say that a boundary point of B is not an interior point of

as your two posts seem to point to, again, I would ask, "Why bother saying that?"

I am always very reticent to post in the Analysis forum, because, while I've had many courses in analysis, I find many of the questions to be beyond me at present. I merely point this out for your reference.