Putnam 1995 A4

Suppose we have a necklace of $n$ beads. Each bead is labeled with an integer and the sum of all these labels is $n-1$. Prove that we can cut the necklace to form a string whose consecutive labels $x_{1},x_{2},\dots,x_{n}$ satisfy $$\sum_{i=1}^{k} x_{i} \leq k-1 \text{for} k=1,2,\dots,n.$$