Rearranging small sets for distinct partial sums

A conjecture of Graham (repeated by Erd{o}s) asserts that for any set A ⊆ Fₚ \0\, there is an ordering a₁, …, a|A| of the elements of A such that the partial sums a₁, a₁+a₂, …, a₁+a₂+⋯+a|A| are all distinct. We give a very short proof of this conjecture for sets A of size at most log p/loglog p.