2001 IMO Shortlist S02

Let $a_0, a_1, a_2, \ldots$ be an arbitrary infinite sequence of positive numbers. Show that the inequality $1 + a_n > a_{n-1} \sqrt[n]{2}$ holds for infinitely many positive integers $n$ .

令 $a_0, a_1, a_2, \ldots$ 为任意无限正数序列。证明不等式 $1 + a_n > a_{n-1} \sqrt[n]{2}$ 对于无穷多个正整数 $n$ 成立。