But in this case the right side is exactly that, the sum of an infinite sequence of numbers (which equals the limit of the partial sum for that sequence if it c

But in this case the right side is exactly that, the sum of an infinite sequence of numbers (which equals the limit of the partial sum for that sequence if it converges (which this sequence of partial sums does)). The reason it still works is because it's actually 0 < s_0 < a < s_1 < 1, so even if a approaches s_1 arbitrarily close it still won't be equal to an integer (same is true if it goes arbitrarily close to s_0 of course).