## Asymptotic characterization in terms of sequence limits

Suppose is a continuous function such that for every we have

Prove that .

*Solution:* In problems like this Baire’s theorem almost suggests itself. For some define the following sets

and note that the hypothesis implies that for every there exists such that . This proves that . Moreover, from the definition we have for . The continuity of assures us that the sets are closed.

The Baire category theorem implies that at least one has non-void interior and that means that there exists a non-trivial interval such that

The main idea now is to realize that contains every real number which is sufficiently large. To do this, suppose with and note that the inequality can only hold for a finite number of values for .

Thus we have proved that for there exists such that for every . This is exactly the definition of the fact that the limit of at is zero.