I suppose I'd say that, for all integers n > 0,[tex]AB = \left( \prod_{i=1}^n (A^{1/2^i} + B^{1/2^i}) \right) \cdot (A^{1/2^n}  B^{1/2^n})[/tex]which is pretty much what you wrote  before trying to produce a limit. The statement is true for all given n, but no limit needs to be involved, as I see it.
For another example, all of the products 1.(1/1), 2.(1/2), 4.(1/4), 8.(1/8), ... equal 1; the fraction is getting smaller and smaller, but that doesn't mean that the sequence 1,2,4,8,... converges, let alone to 1.
