A. 552. Prove that for an arbitrary sequence of nonnegative real numbers and >0 there exist infinitely many positive integers n for which .
(Schweitzercompetition, 2011)
(5 points)
Deadline expired on 10 February 2012.
Solution. We prove by contradiction. Suppose that there is a sequence and >0 for which
 (1) 
holds with finitely many exception. As a simple corollary of (1) we have
it follows that a_{n}>0 and a_{n1}<1. Moreover, from the AMGM inequality we get
so a_{n}>a_{n1}; the sequence increases from the index n_{0}.
By the monotonicity and boundedness the sequence converges. We show that its limit is . Let ; from (2) we see that
(2A1)^{2}0,
therefore . Hence, .
Define . This sequence decreases from the index n_{0}, and it converges to 0. Transforming (1), for n>n_{0} we have
 (3) 
Summing up (3), for n>n_{0} we get a lower bound for x_{n}:
 (4) 
In the rest of the solution we consider the properties of the sequence (n+1)x_{n}. We distinguish two cases.
Case 1: the sequence (n+1)x_{n} has a finite accumulation point. Since except for finitely many indices, all accumulation points lie in the interval . It is wellknown that there is a minimal accumulation point; this point is called as the limes inferior of the sequence. Denote by c the smallest accumulation point. Then there is a sequence of indices such that .
Since there is no smallest accumulation point than c, for all >0, by the BolzanoWeierstrass theorem, (n+1)x_{n}>c holds except for finitely many indices. Substuting this into (3) and summing up again, we obtain that for n_{k}>n_{0} and c,
From the transition k we get
Then from +0, we obtain
contradiction.
Case 2: The sequence (n+1)x_{n} has no accumulation point. By the BolzanoWeierstrass theorem and the positivity this means that for all real K, (n+1)x_{n} with finitely many exceptions, so (n+1)x_{n}. Then there is an index n>n_{0} such that 1<nx_{n1}<(n+1)x_{n}. Applyinig (3),
we get a contradiction, since both nx_{n1}(n+1)x_{n} and 12(n+1)x_{n} are negative.
Remark. The term in the problem statement is important. Ommiting this quantity the statement would be false. For example, for the sequence ,
Statistics on problem A. 552.  

Problems in Mathematics of KöMaL, January 2012
