M361K (56225) Problem Set 1 Solutions
1. Suppose I = . Let cfw_Ai iI be a nonempty family of subsets of a set B . Prove De
Morgans Laws, i.e.,
B\
(B \ Ai )
i I
B\
(1)
(B \ Ai )
Ai =
(2)
i I
Ai =
i I
i I
Solution:
1. First well prove equation (1). Suppose
M361K (56225) Problem Set 1 Solutions
1. Suppose I = . Let cfw_Ai iI be a nonempty family of subsets of a set B . Prove De
Morgans Laws, i.e.,
B\
(B \ Ai )
i I
B\
(1)
(B \ Ai )
Ai =
(2)
i I
Ai =
i I
i I
Solution:
1. First well prove equation (1). Suppose
M361K (56225) Final Exam Solutions
1. (a) (5 points) State Hlders inequality.
o
Solution: Suppose a < b. Let p, q > 1 so that 1/p +1/q = 1. If f, g : [a, b] R
are Riemann integrable, then f g 1 f p g q . Here
1/p
b
f
p
p
|f (x)| dx
=
.
a
(b) (5 points) St
M361K (56225) Final Exam
Name (print):
EID:
Question:
1
2
3
4
5
6
7
Total
Points:
35
10
10
10
10
15
10
100
Score:
This exam is 6 pages long. There are 7 questions. Answer all questions in the space
provided. Phones, calculators, notes, and other aids are
M361K (56225) Final exam practice problems
1. Let f : [a, b] R be a function and p 1. Dene f
p.
2. State the Cauchy-Schwarz inequality.
3. State and prove Youngs inequality.
4. State Hlders inequality.
o
5. State Minkowskis inequality.
6. State Jensens in
M361K (56225) Midterm 2 practice problems
1. Show that the series
2. Show that
n=0
n=2
1/(n log n) diverges.
1/(n + 1)(n + 2) = 1.
3. Suppose that cfw_an n0 is a sequence in R. Let = lim supn
.
(a) Prove the Cauchy-Hadamard theorem: show the series
1 and
M361K (56225) Midterm 1 practice problems
1. Let A and B be sets. Show that
(A B ) \ (A B ) = (A \ B ) (B \ A).
2. Show that the set
Nk
S=
kN
is countable, where
Nk = N N .
k times
3. Show that the set of all functions N N is not nite or countable.
4. Sho
M361K (56225) Final Exam Solutions
1. (a) (5 points) State Hlders inequality.
o
Solution: Suppose a < b. Let p, q > 1 so that 1/p +1/q = 1. If f, g : [a, b] R
are Riemann integrable, then f g 1 f p g q . Here
1/p
b
f
p
p
|f (x)| dx
=
.
a
(b) (5 points) St
M361K (56225) Midterm 2
Name (print):
EID:
This exam is 4 pages long. There are 5 questions. Answer all questions in the space
provided. Phones, calculators, notes, and other aids are forbidden.
1. (10 points) Show that
n=0
1/(n + 1)(n + 2) = 1.
2. (10 po
M361K (56225) Midterm 1
Name (print):
EID:
This exam is 4 pages long. There are 5 questions. Answer all questions in the space
provided. Phones, calculators, notes, and other aids are forbidden.
1. (a) (5 points) State the Cauchy-Schwarz inequality.
(b) (
M361K (56225) Final Exam
Name (print):
EID:
Question:
1
2
3
4
5
6
7
Total
Points:
35
10
10
10
10
15
10
100
Score:
This exam is 6 pages long. There are 7 questions. Answer all questions in the space
provided. Phones, calculators, notes, and other aids are
M361K (56225) Problem Set 2 Solutions
1. (a) Let x Rn . Dene x = max1in |xi |. Show that for all x, y Rn , we
have x + y x + y .
Solution: Let x, y Rn . If 1 j n, then |xj | max1in |xi | and |yj |
max1in |yi |, so
|xj + yj | |xj | + |yj | max |xi | + max
M361K (56225) Problem Set 3 Solutions
1. Suppose that U = (a, b) with a < b. Recall that f : U R is -Hlder continuous if
o
there exists (0, 1] and C > 0 so that for all x, y U , the inequality |f (x) f (y )|
C |x y | holds. Show that if there exists a co
M361K (56225) Problem Set 2 Solutions
1. (a) Let x Rn . Dene x = max1in |xi |. Show that for all x, y Rn , we
have x + y x + y .
Solution: Let x, y Rn . If 1 j n, then |xj | max1in |xi | and |yj |
max1in |yi |, so
|xj + yj | |xj | + |yj | max |xi | + max
M361K (56225) Problem Set 3 Solutions
1. Suppose that U = (a, b) with a < b. Recall that f : U R is -Hlder continuous if
o
there exists (0, 1] and C > 0 so that for all x, y U , the inequality |f (x) f (y )|
C |x y | holds. Show that if there exists a co
M361K (56225) Problem Set 4 Solutions
1. The gamma function is dened for s > 0 by the improper integral
1/
xs1 ex dx = lim
(s) =
0
>0
0
xs1 ex dx.
(a) Show that the integral in the denition (s) converges when s > 0.
Solution: The integrand xs1 ex is nonne
M361K (56225) Problem Set 5 Solutions
1. (Rudin) Let X be a set. Suppose cfw_fn and cfw_gn are sequences of functions X R
converging uniformly to f and g respectively.
(a) Show that cfw_fn + gn converges uniformly to f + g .
Solution: Let > 0. There ex
M361K (56225) Midterm 1 Solutions
1. (a) (5 points) State the Cauchy-Schwarz inequality.
Solution: Let n > 0 and let x, y Rn . Then
|x y | x
y,
i.e.,
n
n
xi y i
i=1
1/2
n
x2
i
i=1
1/2
2
yi
.
i=1
(b) (5 points) Let (X, d) be a metric space. State the tria
M361K (56225) Midterm 1 Solutions
1. (a) (5 points) State the Cauchy-Schwarz inequality.
Solution: Let n > 0 and let x, y Rn . Then
|x y | x
y,
i.e.,
n
n
xi y i
i=1
1/2
n
x2
i
i=1
1/2
2
yi
.
i=1
(b) (5 points) Let (X, d) be a metric space. State the tria
M361K (56225) Problem Set 5 Solutions
1. (Rudin) Let X be a set. Suppose cfw_fn and cfw_gn are sequences of functions X R
converging uniformly to f and g respectively.
(a) Show that cfw_fn + gn converges uniformly to f + g .
Solution: Let > 0. There ex
M361K (56225) Problem Set 4 Solutions
1. The gamma function is dened for s > 0 by the improper integral
1/
xs1 ex dx = lim
(s) =
0
>0
0
xs1 ex dx.
(a) Show that the integral in the denition (s) converges when s > 0.
Solution: The integrand xs1 ex is nonne
M361K (56225) Final Exam Solutions
1. (a) (5 points) State Hlders inequality.
o
Solution: Suppose a < b. Let p, q > 1 so that 1/p +1/q = 1. If f, g : [a, b] R
are Riemann integrable, then f g 1 f p g q . Here
1/p
b
f
p
p
|f (x)| dx
=
.
a
(b) (5 points) St