Problem

You have 1/2 < a < 1 and the infinite sequence a_0, a_1, a_2 ...
with a_i = a^i.

Can you construct sets S_n, T_n such that for all i < n either a_{2i} \elem S and a_{2i+1} \elem T or a_{2i} \elem T and a_{2i+1} holds, with lim n -> \infty S_n - T_n = 0?
That is to say, the sum of the elements in S_{\infty} - the sum of the elements in T_{\infty} = 0.

I'm pretty sure.  To start, put a_1 in S and a_2 in T.  Then step by step assign the larger of each subsequent pair to whichever set has the smaller cumulative sum.

cba to prove it right now, but I think that'll work.