2001 IMO Shortlist Problems/A2
Let be an arbitrary infinite sequence of positive numbers. Show that the inequality holds for infinitely many positive integers .
We proceed with a proof by contradiction. Suppose the statement were false. Then, there exists a sequence of positive integers for which there are only finitely many with . Let the largest such be , so that whenever . Then, it is clear that for all nonnegative . Therefore, define . If there does not exist a sequence of positive integers for which , it is clear that there will not exist any sequence for which that property is eventually true.
Thus, I claim there does not exist a sequence of positive integers for which . Again, suppose there does exist such a sequence. Then, define and . It is clear that for all . I claim that this sequence will always become eventually negative. Note that , which becomes negative if and only if does. In other words, becomes zero if is unbounded. However, is eventually less than , so this sum is indeed unbounded and the proof is complete.