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 series
converges uniformly on
.