IR-MO-R2 2013 Problem 2

Let $n$ be a natural number and suppose that $w_1, w_2, \ldots , w_n$ are $n$ weights . We call the set of $\{ w_1, w_2, \ldots , w_n\}$ to be a Perfect Setif we can achieve all of the $1,2, \ldots, W$ weights with sums of $w_1, w_2, \ldots , w_n$ , where $W=\sum_{i=1}^n w_i$ . Prove that if we delete the maximum weight of a Perfect Set, the other weights make again a Perfect Set.