SCHWEITZER 2007 Problem 4

Let $p$ be a prime number and $a_1, \ldots, a_{p-1}$ be not necessarily distinct nonzero elements of the $p$ -element $\mathbb Z_p \pmod{p}$ group. Prove that each element of $\mathbb Z_p$ equals a sum of some of the $a_i$ 's (the empty sum is $0$ ).

(translated by Miklós Maróti)