HOMEWORK #3 extra-credit
Math 5615H, section 1
due Sunday, September 30
Solutions are to be submitted separately (preferably by email to dbilyk@umn.edu, typed, scanned, or
photographed in readable quality). Partial credit just for eort will not be given p
HOMEWORK #2: solutions
Math 5615H, section 1
1. Show that the multiplication axioms (M1M5, see page 5) hold in the complex eld C.
Proved on page 13 of the textbook.
2. Problem # 8, page 22.
Prove that no order can be dened in the complex eld which turns i
HOMEWORK #1
Math 5615H, section 1
Solutions
1. a) Problem #1, page 21.
Solution: Let us write the rational number r as r = p , where p, q Z.
q
1) Assume r + x Q, then r + x =
m
,
n
x = (r + x) r =
m, n Z. Therefore,
m p
mq pn
=
Q,
n
q
nq
since both the
Math 5615H
Homework 11
Posted: 11/22; Updated: 11/24; due: Monday, 12/1/2014
The problem set is due at the beginning of the class on Monday after
the Thanksgiving Break.
Reading: Chapter 5: pages 109-113.
For this homework, you may assume that ( x) = 1/(2
Appendix A. Exponential and Logarithmic Functions
For fixed b > 1, the function bx was defined in Exercise 6 on p.22 in the textbook "Principles of Mathematical Analysis" by W. Rudin. It satisfies bx > 0, and (E1). bx+y = bx by for real x, y. In particula
Math 5615H. Name (Print)
November 16, 2011.
2nd Midterm Exam.
60 points are distributed between 5 problems. You have 50 minutes (2:30 pm 3:20 pm) to work on these problems. No books, no notes, except for: Appendix A. Exponential and Logarithmic Functions
Math 5615H. Name (Print)
October 5, 2011.
Midterm Exam.
60 points are distributed between 5 problems. You have 50 minutes (2:30 pm 3:20 pm) to work on these problems. No books, no notes. Calculators are permitted, however, for full credit, you need to sho
Math 5615H: Introduction to Analysis I. Syllabus: Fall 2011
(updated on September 8) Class Times and Location: 2:30 pm 3:20 pm MWF, VinH 211. Instructor: Mikhail Safonov, VinH 231, tel: 625-8571, email: safonov@math.umn.edu http:/www.math.umn.edu/safonov
Math 5615H. November 16, 2011. Midterm Exam 2. Problems and Solutions. Problem 1. (10 points). Let A1 , A2 , A3 , . . . be subsets of a metric space (X, d).
If B :=
k=1
Ak ,
prove that B
k=1
Ak ,
where B and Ak denote the closures of B and Ak . Show, b
Math 5615H. October 5, 2011. Midterm Exam 1. Problems and Solutions. Problem 1. (10 points). Let A and B be nonempty bounded subsets of R. Show that sup(A B) = supcfw_sup A, sup B. Proof. Denote M1 := sup(A B), M2 := supcfw_sup A, sup B. We have x sup A M
Math 5615H: Introduction to Analysis I. Homework #13. Problems and Solutions.
Fall 2011
#1. If f is a continuous mapping of a metric space X into a metric space Y , prove that f (E) f (E) for every set E X. (E denotes the closure of E). Show, by an exampl
Fixed point theorem for contractions
Denition
Let X be a metric space and T : X X be a function. We say
that T is a contraction if there exists 0 < q < 1 such that
d T (x), T (y) q d(x, y)
for each x, y X.
Dmitriy Bilyk
Fixed point theorem
Fixed point the
Vector spaces
Rn (Cn ) a set of all ordered n-tuples x = x1 , ., xn ,
where xk R (C) with operations:
Addition:
x + y = x1 + y1 , ., xn + yn
Multiplication by a scalar : for Rn (C):
x = (x1 , x2 , ., xn )
Dmitriy Bilyk
CauchySchwarz inequality
Vector spac
HOMEWORK #3
Math 5615H, section 1
due Friday, September 28
(1) Let a1 , ., an be real numbers. Prove that
n
a2
k
k=1
n
k=1 |ak |
n
.
(2) Let a1 , ., an be real numbers. Prove that
n
n
k=1
n
|ak |4/3 .
|ak |2/3
ak
k=1
k=1
(3) Prove that for all 2-dimens
Functions
Let A and B be sets
Dmitriy Bilyk
Finite & Countable sets
Functions
Let A and B be sets
A function (a mapping) f from A (in)to B: a rule that
associates to each element x A an element f (x) B.
Dmitriy Bilyk
Finite & Countable sets
Functions
Let
HOMEWORK #2
Math 5615H, section 1
due Friday, September 21
1. Show that the multiplication axioms (M1M5, see page 5) hold in the complex eld C.
2. Problem # 8, page 22.
3. Problem # 13, page 23.
4. Problem # 14, page 23.
5. Problem # 17, page 23.
6. Write
HOMEWORK #1
Math 5615H, section 1
due Friday, September 14
1. a) Problem #1, page 21. b) If both x and y are irrational, do x + y or x + y have to
be irrational?
2. Prove that 2 + 7 is irrational.
3. Problem # 3, page 22 (i.e. prove Proposition 1.15 on pa
Unions and cartesian products
Theorem
Let cfw_En , n N be a sequence of countable sets. Then
En
n=1
is countable.
Dmitriy Bilyk
Countable & uncountable sets
Unions and cartesian products
Theorem
Let cfw_En , n N be a sequence of countable sets. Then
En
n=
HOMEWORK #5
Math 5615H, section 1
due Friday, October 12
(1) Let cfw_xn be a convergent sequence of nonnegative numbers and let lim xn = x. Prove that
n
lim xn = x.
n
Proof: Case (a) x = 0: Take any > 0. Then there exists N N such that for all n N we hav
Some practice problems
Math 5615H, section 1
In no particular order:
(1) Prove Lagranges identity:
n
2
a jb j
j=1
n
n
=
a2
j
b2
k
j=1
k=1
1
2
n
n
(a j bk ak b j )2 .
j=1 k=1
Give a geometric interpretation of this identity in the case n = 3.
Use the ide
Math 5615H, Section 001
Dmitriy Bilyk, Fall 2012
Midterm 1, October 15, 2012
Time Limit: 50 minutes
Name (Print):
Student ID:
Signature:
This exam contains 5 pages (including this cover page) and 6 problems. Check to see if any pages
are missing. Enter al
HOMEWORK #3 - SOLUTIONS
Math 5615H, section 1
due Friday, September 28
(1) Let a1 , ., an be real numbers. Prove that
n
n
k=1 |ak |
a2
k
n
k=1
.
Solution: Consider the real numbers bk , k = 1, ., n, dened by bk = 1 if ak 0 and bk = 1 if
ak < 0. Then n b2
HOMEWORK #4 - SOLUTIONS
Math 5615H, section 1
due Friday, October 5
CHAPTER 2:
Problem 9: Let E denote the set of interior points of a set E. [E is called the interior of E.]
(a) Prove that E is always open.
Proof: If x E , then there is a neighborhood B
Math 5615H: Introduction to Analysis I. Homework #12. Problems and Solutions.
Fall 2011
#1. Let aj and bj,k be real numbers defined for all j, k = 0, 1, 2, . . ., such that
|aj | A = const < ,
j=0
|bj,n | B = const < for all j, n;
and lim bj,n = 0 for eac
Math 5615H: Introduction to Analysis I. Homework #11. Problems and Solutions. #1. (12 points). Find
n S := . 2n n=1
Fall 2011
Solution. Substituting n = m + 1, we get
n m+1 ( 1 )m 2S = = =S+ = S + 2, 2n-1 m=0 2m 2 n=1 m=0
so that S = 2. #2. (10 points).
Math 5615H: Introduction to Analysis I.
Fall 2011
Homework #11 (due on Wednesday, November 23). 50 points are divided between 4 problems. #1. (12 points). Find S :=
n=1
n . 2n
Hint. One can try to rewrite 2S in terms of S, or use Theorem 3.41 with an = 1
Math 5615H: Introduction to Analysis I.
Fall 2011
Homework #10 (due on Wednesday, November 16). 50 points are divided between 4 problems. #1. (12 points). The sequence cfw_an is defined by a1 = 0, and an+1 = 2an /2 for n = 1, 2, 3, . . . .
Prove that cfw
Math 5615H: Introduction to Analysis I.
Fall 2011
Homework #9 (due on Wednesday, November 9). 50 points are divided between 4 problems. #1. (10 points). Using partial-fraction decomposition 1 A B C = + + , n(n + 1)(n + 2) n n+1 n+2 show that the series
n
Math 5615H: Introduction to Analysis I.
Fall 2011
Homework #8 (due on Wednesday, November 2). 50 points are divided between 4 problems. #1. (12 points). Let 0 < x1 = a < x2 = b be arbitrary real number, and let 1 xn := (xn-2 + xn-1 ) for n = 3, 4, 5, . .
Math 5615H: Introduction to Analysis I.
Fall 2011
Homework #7 (due on Wednesday, October 26). 50 points are divided between 4 problems. You can use the following Theorem which was proved in class. Theorem. A subset K of a metric space (X, d) is compact in