1975 IMO 第 4 题

Let $a_{1}, a_{2}, \ldots, a_{n}, \ldots$ be a sequence of real numbers such that $0 \leq a_{n} \leq 1$ and $a_{n}-2 a_{n+1}+a_{n+2} \geq 0$ for $n=1,2,3, \ldots$. Prove that $$0 \leq(n+1)\left(a_{n}-a_{n+1}\right) \leq 2 \quad \text { for } n=1,2,3, \ldots$$

令 $a_{1}, a_{2}, \ldots, a_{n}, \ldots$ 为实数序列，使得 $0 \leq a_{n} \leq 1$ 和 $a_{n}-2 a_{n+1}+a_{n+2} \geq 0$ 对于 $n=1,2,3, \ldots$。证明 $$0 \leq(n+1)\left(a_{n}-a_{n+1}\right) \leq 2 \quad \text { for } n=1,2,3, \ldots$$