2007 CMO 第 5 题

Let $\{a_n\}_{n \geq 1}$ be a bounded sequence satisfying

$$a_n < \displaystyle\sum_{k=a}^{2n+2006} \frac{a_k}{k+1} + \frac{1}{2n+2007} \quad \forall \quad n = 1, 2, 3, \ldots$$

Show that

$$a_n < \frac{1}{n} \quad \forall \quad n = 1, 2, 3, \ldots$$

令 $\{a_n\}_{n \geq 1}$ 为有界序列，满足

$$a_n < \displaystyle\sum_{k=a}^{2n+2006} \frac{a_k}{k+1} + \frac{1}{2n+2007} \quad \forall \quad n = 1, 2, 3, \ldots$$

表明

$$a_n < \frac{1}{n} \quad \forall \quad n = 1, 2, 3, \ldots$$