The issue is that the precision of the square roots needs to increase in order to guarantee a result. Consider an algorithm to generate the worst case scenario

The issue is that the precision of the square roots needs to increase in order to guarantee a result. Consider an algorithm to generate the worst case scenario that starts with two lists that have sqrt sums that differ by D. Append to the lists numbers x and y such that |sqrt(x) - sqrt(y)| > D and append them so that the previously lesser list is now greater but also the absolute difference decreases each step.