Introduction to Analysis
Homework 11
1. (Rudin ex7.11) Suppose {fn},{gn} are defined on E, and (a) P
fn has uniformly bounded partial sums;
(b) gn →0 uniformly on E;
(c) g1(x)≥g2(x)≥g3(x)≥ · · · for every x∈E.
Prove that P
fngn converges uniformly on E. Hint: Compare with Theorem 3.42
2. (Rudin ex7.12) Suppose g and fn(n = 1,2,3,· · ·) are defined on (0,∞), are Riemann- integrable on [t, T] whenever 0 < t < T < ∞, |fn| ≤ g, fn → f uniformly on every compact subset of (0,∞), and
Z ∞
0
g(x)dx <∞.
Prove that
n→∞lim Z ∞
0
fn(x)dx= Z ∞
0
f(x)dx.
3. (Rudin ex7.13) Assume that {fn} is a sequence of monotonically increasing functions on R1 with 0≤fn(x)≤1 for all x and all n.
(a) Prove that there is a functionf and a sequence{nk} such that f(x) = lim
k→∞fnk(x) for every x∈R1.
(b) If, moreover, f is continuous, prove that fnk →f uniformly on R1. (There are hints in the book. I skip them here.)
4. (Rudin ex7.24) Let X be a metric space, with metric d. Fix a point a ∈ X. Assign to each p∈X the function fp defined by
fp(x) = d(x, p)−d(x, a) (x∈X).
Prove that |fp(x)| ≤d(a, p) for all x∈X, and that thereforefp ∈ C(X).
Prove that
kfp −fq k=d(p, q) for all p, q ∈X.
If Φ(p) = fp it follows that Φ is an isometry (a distance-preserving mapping) of X onto Φ(X)⊂ C(X).
Let Y be the closure ofφ(X) in C(X). Show that Y is complete.
5. If{fn}is a sequence of differentiable functions on [a, b] such that the sequence of deriva- tives fn0 converges uniformly, then the sequence of functions gn defined by gn(x) = fn(x)−fn(a) converges uniformly.
6. Suppose that the series P∞
n=0cn(x−a)n converges for x = x1 with x1 6= a. Then the series converges uniformly on I = {x|a−h ≤ x ≤ a+h} for each h < |x1 −a|. Also, there is a number M such that
|cn(x−a)n| ≤M · h
|x1−a|
n
for
|x−a| ≤h <|x1−a|.
7. Given that P∞
n=1n|bn| converges. Let f(x) = P∞
n=1bnsinnx. Show that f0(x) = P∞
n=1nbncosnx and that both series converges uniformly for allx.
