So there exists a sequence {x_n} in A \/ B s.t. x_n --> x . For any n, x_n belongs either to A or to B. It can't be that both A and B contain a finite number of elements of the sequence ==> either A or B (or both, of course, as "or" in mathematics is always inclusive unless otherwise stated) contain an infinite number of elements of {x_n} ==> since {x_n} converges so does any infinite subsequence and to the same limit ==> we're done.

Tonio