ILL 1976 Problem 24

Let $0 \le x_1 \le x_2\le\cdots\le x_n \le 1$ . Prove that for all $A \ge 1$ , there exists an interval $I$ of length $2\sqrt[n]{A}$ such that for all $x \in I$ , $$|(x - x_1)(x - x_2) \cdots (x -x_n)| \le A.$$