IMO 2005 Problem 2

Let $a_1,a_2,\ldots$ be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer $n$ the numbers $a_1,a_2,\ldots,a_n$ leave $n$ different remainders upon division by $n$ .

Prove that every integer occurs exactly once in the sequence $a_1,a_2,\ldots$ .