 Some have felt that restricting additivity to merely countable sums is arbitrary, and is merely an artifact of the summation technique introduced in section  3.2. An alternate technique for summing infinite sets of non-negative numbers takes advantage of the fact that a sum of nonnegative numbers as defined earlier doesn’t depend on the order of the terms. We consider the set of all partial sums of arbitrary finite subsets of the set, and take the least upper bound of this set to represent the sum of the set as a whole. If this sum is some positive finite value \(k\), then we can see that at most \(nk\) of the terms in the set being summed can be greater than 1/\(n\). Since every positive real number is greater than 1/\(n\) for some \(n\), this means that the set of positive elements of the set is a countable union of finite sets, and thus must be countable. That is, if a set being summed in this way has uncountably many non-zero elements, the sum must be infinite.