2001 IMO Shortlist Problems/A2
Problem
Let be an arbitrary infinite sequence of positive numbers. Show that the inequality
holds for infinitely many positive integers
.
Solution
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.