A zero-sum theorem over Z

A zero-sum sequence of integers is a sequence of nonzero terms that sum to 0. Let $k>0$ be an integer and let $[-k,k]$ denote the set of all nonzero integers between $-k$ and $k$. Let $\mathrm{\ell }(k)$ be the smallest integer $\mathrm{\ell }$ such that any zero-sum sequence with elements from $[-k,k]$ and length greater than $\mathrm{\ell }$ contains a proper nonempty zero-sum subsequence. In this paper, we prove a more general result which implies that $\mathrm{\ell }(k)=2k-1$ for $k>1$.