Can anyone suggest a reference or a simple proof of the following?

For every sequence of integers $a_1,...,a_n$, there exists a non-empty subsequence whose sum is divisible by $n.$

Here is a more general problem: Prove that for every $r$ and $n$ there is $k$ such that for every sequence of vectors $v_1,...,v_k \in {\mathbb Z}^r$ there exists a subsequence whose sum belongs to $(n{\mathbb Z})^r$. (I would think that $k=n^r$ should be enough...)