Analysis: Sets A & B - Does B Contain a Limit Point of A?

[zilla]

Homework Statement

Suppose that each of A and B is a set such that:
(a) A is a subset of [0,1], B is a subset of [0,1]
(b) Neither of A or B is empty
(c) 0 (zero) is an element of A
(d) The union of A & B = [0,1]
(e) A & B are disjoint
(f) A contains no limit points of B.

Then, B contains a limit point of A.

Homework Equations

None

The Attempt at a Solution

Not sure where to go with it...

 

[ystael]

If $$A$$ contains no limit point of $$B$$, then what kind of points does $$A$$ consist of? (What must $$A$$ look like near $$0$$? How far from $$0$$ can you "push" this appearance?)