Let
![a_0, a_1, a_2, \ldots](/media/m/e/2/6/e2656939bc7d2ee3e6727f4bbc1fcddc.png)
be an arbitrary infinite sequence of positive numbers. Show that the inequality
![1 + a_n > a_{n-1} \sqrt[n]{2}](/media/m/6/8/3/68300d4c59f2a509af66424e1feb09f2.png)
holds for infinitely many positive integers
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
.
%V0
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 $n$.