1549901369-Elements_of_Real_Analysis__Denlinger_

9.2 Uniform Convergence 555

llgkll = fc ---+ 0, so gk ---+ 0 uniformly on [o, ?T - o]. Therefore, by Dirichlet's

-^00 sinkx
test for umform convergence, I: -k-converges uniformly on [o, ?T - o].
k=l

(^00) sin kx
(b) The series I: - k- converges pointwise everywhere, as we showed in
k=l
Example 8.5.7. To investigate uniform convergence we use the uniform Cauchy

. n sinkx
criterion. Consider the partial sums Sn= I: -k-. Let m EN. Note that for
k=l
m:::; k:::; 2m, ~:::; 2 ";,,:::; 1, so sin 2 ";,, ~sin~. Now, let x = 2 !n. Then,
2m. k 2m. k 2m. 1. 1 2m. 1
L
---sm x L sin --^2 m > L -----sm^2 sm^2 L 1---_ sm^2.
k k - 2m 2m 2
k=m k=m k=m k = m

sin l^00 sin kx
Therefore, llS2m - Smll ~ --^2. So, by the uniform Cauchy criterion, I: -k-
2 k=l
cannot converge uniformly on [O, ?r]. 0

EXERCISE SET 9.2

1. Prove Theorem 9.2.2.


2. Prove that if Ji, f2, · · · , fn E B(S), t hen Ji+ f2 + · · · + fn E B(S), and
   llJi + f2 + · · · + fnll:::; llJill + llf211 + · · · + llfnll·


3. Write out the c:-no definition of the statement that f n ---+ f pointwise on a
   set S. Explain how this differs from the definition of uniform convergence
   off n to f on S given in Definition 9.2.3.


4. Which of the following are equivalent? Explain the relevance to Definition
   9.2.3.
   (a) Ve:> 0, :3 no EN 3 Vn ~no, x E S =? lfn(x) - f(x)I < c:.
   (b) Ve:> 0, :3 no EN 3 Vn ~no and Vx E S , lfn(x) - f(x)I < c:.
   (c) Ve:> 0, :3 no EN 3 Vx E Sand Vn ~no, lfn(x) - f(x)I < c:.
   (d) Ve:> 0, :3 no EN 3 Vx E S , n ~no=? lfn(x) - f(x)I < c:.


5. For each of the sequences in Examples 9.1.7 (b) and (d), calculate llfn - !II
   and use it to show that {f n} does not converge uniformly to f.


6. Prove Theorem 9.2.5.


7. Prove that {sin (x + ~)} converges uniformly to sinx on JR. [Hint: use
   the mean value theorem.]