Assume for the sake of contradiction that no nonempty zero-sum subset exists.

To use pigeonhole modulo (which is really the only reasonable way), we need to get both and involved. The easiest way to do this is to WLOG let be pairwise distinct (we know such numbers exist by the problem hypothesis) and consider the partial sums (which are pairwise distinct and nonzero by our assumption; we start at the partial sum so that when we subtract elements from we still have a sum of distinct ).

Now we need another set of at least nonzero elements for pigeonhole to work; something like quickly comes to mind (since the are distinct, at most one of them can satisfy , and we can afford to lose a few), provided we make it compatible with . But this is easy, since adding to every entry to get does not make it less compatible with .

WLOG for (we can assign the roles to and if necessary); then the sets , , and each contain pairwise distinct nonzero elements modulo . But they have elements in all, so by pigeonhole some two sets must intersect (we are only dealing with the nonzero elements). If the third set intersects with either the first or the second, we obviously get a nonempty zero-sum subset; otherwise, if the first two sets intersect, then we must have and for some , which also gets a nonempty zero-sum subset since contains at least terms.

For a real number , let denote how far is from its closest integer multiple of (i.e. absolute value ). If we go by contradiction, then we have that , where represent the positions of on an angular scale. To utilize this local information (which restricts what points can be close to each other) in the context of our global situation of reals spread out , we want to get a lot of the in some interval .

On average, contains of the . If some contains or more of the , then two of the indices must satisfy , contradiction. Otherwise, the interval contains exactly of the for almost all (except for a set of zero density), which is impossible as we consider for any and sufficiently small unless for some ; but then we get or more in .