# How to prove greedy algorithm for number partitioning?

the partition problem (or number partitioning1) is the task of deciding whether a given multiset S of positive integers can be partitioned into two subsets S1 and S2 such that the sum of the numbers in S1 equals the sum of the numbers in S2.

There is a greedy algorithm for this problem:

One approach to the problem, imitating the way children choose teams for a game, is the greedy algorithm, which iterates through the numbers in descending order, assigning each of them to whichever subset has the smaller sum. This approach has a running time of O(n log n). This heuristic works well in practice when the numbers in the set are of about the same size as its cardinality or less, but it is not guaranteed to produce the best possible partition. For example, given the set S = {4, 5, 6, 7, 8} as input, this greedy algorithm would partition S into subsets {4, 5, 8} and {6, 7}; however, S has an exactly balanced partition into subsets {7, 8} and {4, 5, 6}.

But, I don't know how to prove "This heuristic works well in practice when the numbers in the set are of about the same size as its cardinality or less". Can anyone help?