# 2002 IMO Shortlist Problems/A2

## Problem

Let be an infinite sequence of real numbers, for which there exists a real number with for all , such that

Prove that .

## Solutions

### Solution 1

For some fixed value of , let be the permutation of the first natural numbers such that is an increasing sequence. Then we have

Now, by the Cauchy-Schwarz Inequality, we have

Thus for all , we must have

and therefore must be at least 1, Q.E.D.

### Solution 2

We proceed to as in Solution 1. We now note that by the AM-HM Inequality,

Thus for any , we have two that differ by more than . But this becomes arbitrarily close to 1 as becomes arbitrarily large. Hence if we had , then we could obtain a contradiction, so we conclude that , Q.E.D.

*Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.*

## Notes

The chief difficulty of this problem seems to be obtaining ; once this result has been obtained, there are many ways to conclude.