Let be a decreasing sequence. Prove that converges uniformly if-f .

**Solution**

Since then for a given there exists positive integer such that if to hold

(1)

It suffices to prove the result for due to symmetry and periodicity. So, it suffices to prove that for it holds for all and all .

We distinguish cases:

- ;
- ;
where .

This completes the proof of the first part. Now the converse is much easier. For each such that we have . Hence . Therefore, picking we have:

(2)

Since the series converges uniformly we have that uniformly. More specifically, as . Using the sandwich theorem it follows that . It only remains to prove that . This follows from

Exercise:Prove that the seriesconverges uniformly on .