A simple proof of a theorem of Bollobás and Leader

By using Scherk's lemma we give a simple combinatorial proof of a theorem due to Bollobás and Leader. For any sequence of elements of an abelian group of order $k$, calling the sum of $k$ terms of the sequence a $k$-sum, if 0 is not a $k$-sum, then there are at least $r-k+1$ $k$-sums.