Student Information
Intro to modern analysis I, Math S3027D (Section 2)
May 25, 2010
Name:
Major(s):
Email Address:
School:
I would like to make a course webpage with the information above listed. The
idea is to help students work together in this class.
Modern Analysis, Homework 1
Possible solutions
1. (1 pt) Let k be a eld and x k be non-zero. Show that 1/(1/x) = x.
Solution: First of all, notice that multiplicative inverses are unique in elds: if x k \ cfw_0
and y, z k satisfy xy = xz = 1, then
y =1 y
Modern Analysis, Homework 9
Answers available on Aug 10, or sooner
Review problems for chapter 9. Recommended problems are marked with a *. Throughout, is an increasing function on the interval [a, b].
1.* [10] Rudin, Ch 6, #1: Suppose that a x0 b, is con
Modern Analysis, Homework 9
Possible solutions
Review problems for chapter 9. Recommended problems are marked with a *. Throughout, is an increasing function on the interval [a, b], where a < b.
1.* [10] Rudin, Ch 6, #1: Suppose that a x0 b, is continuous
Modern Analysis, Homework 8
Due August 6, 2010
All problems are worth 5 points.
1.
2.
3.
4.
5.
[12]
[18]
[11]
[12]
[13]
Rudin,
Rudin,
Rudin,
Rudin,
Rudin,
Ch
Ch
Ch
Ch
Ch
5,
5,
5,
5,
5,
problem
problem
problem
problem
problem
1.
2.
3.
5.
8.
The following p
Modern Analysis, Homework 8
Possible solutions
All problems are worth 5 points.
1. [12] Rudin, Ch 5, problem 1.
Solution: Since |f (x) f (y )| (x y )2 for all x, y R, if x = y , we have
0
f ( x) f ( y )
|x y |
xy
Letting y x, of course |x y | 0 and 0 0,
Modern Analysis, Homework 7
Due July 29, 2010
1. [12] (5 pts) Rudin, Ch 4, problem 2.
2. [16] (5 pts) Suppose that A and B are connected subsets of a metric space (X, d) and
that A B = cfw_. Prove that A B is connected.
3. [13] (5 pts) Let f : X Y be a ma
Modern Analysis, Homework 7
Possible solutions
1. [12] (5 pts) Rudin, Ch 4, problem 2.
Solution: Let f : X Y be a continuous mapping of metric spaces. Let y f (E ). Then
y = f (e) for some e E = E E . If e E , then y = f (e) f (E ) f (E ). If e E , then
t
Modern Analysis, Homework 6
Due July 15, 2010
1. [15] (5 pts) Rudin, Ch 3, problem 4.
2. [23] (5 pts) Rudin, Ch 3, problem 5.
3. [15] (5 pts) Rudin, Ch 3, problem 6.
4. [20] (5 pts) Rudin, Ch 3, problem 8.
5. [22] (5 pts) Let S9 N be the set of all positi
Modern Analysis, Homework 6
Possible solutions
1. [15] (5 pts) Rudin, Ch 3, problem 4.
1
Solution: First I claim that s2m = 1 2m1 and that s2m+1 = 1 21 . Lets prove this by
m
2
1
induction. Let P (n) be the statement: sn = 1 2n/2 if n is even, or sn = 1 n
Modern Analysis, Homework 5
Due ?, 2010
The following problems are optional, and are worth 5 pts each:
E1. [30] Let P (x) be a degree n polynomial with integral coecients and no rational root.
Suppose that z is a root of P . Prove that there exists A > 0
Modern Analysis, Homework 5
Possible solutions
The following problems are optional, and are worth 5 pts each:
E1. [30] Let P (x) be a degree n polynomial with integral coecients and no rational root.
Suppose that z is a root of P . Prove that there exists
Modern Analysis, Homework 4
Due July 1, 2010
1. [12] (5 points) Prove that every sequence of reals has a monotone (ie either decreasing or
increasing, though not necessarily strictly ) subsequence.
2. [15] (10 points) Rudin, Ch 3, problem 16, parts (a) an
Modern Analysis, Homework 4
Possible solutions
1. [12] (5 points) Prove that every sequence of reals has a monotone (ie either decreasing or increasing, though not necessarily strictly so) subsequence.
Solution: Let cfw_xn be a sequence of reals. Suppose
Modern Analysis, Homework 3
Due June 22, 2010
Directions: hand in all of problems 0-5, but please try problem 6.
0.[12] (2 pts) Rudin, Ch 2, problem 12.
1.[12] (2 pts) Rudin, Ch 2, problem 14.
2.[10] (6 pts) Rudin, Ch 2, problem 9.
3.[25] (5 pts) Rudin, C
Modern Analysis, Homework 3
Possible solutions
0. (2 pts) Rudin, Ch 2, problem 12: Let K R consist of 0 and the numbers 1/n for
n = 1, 2, 3, . Prove that K is compact directly from the denition (without using the
Heine-Borel theorem).
Solution: Let cfw_U
Modern Analysis, Homework 2
Due June 10, 2010
1. (2 pts) Suppose f : A B is a function. Prove that f is a bijection if and only if it is
both injective (one-to-one) and surjective (onto).
2. (2 pts) Prove by induction that for all positive integers n,
n
i
Modern Analysis, Homework 2
Possible solutions
1. (2 pts) Suppose f : A B is a function. Prove that f is a bijection if and only if it is
both injective (one-to-one) and surjective (onto).
Solution: Suppose rst that f is a bijection, ie there exists a fun
Modern Analysis, Homework 1
Due June 3, 2010
1.
2.
3.
4.
5.
6.
(1 pt) Let k be a eld and x k be non-zero. Show that 1/(1/x) = x.
(3 pts) Rudin, Ch 1, problem 5.
(5 pts) Rudin, Ch 1, problem 6.
(10 pts) Rudin, Ch 1, problem 7.
(2 pts) Rudin, Ch 1, problem
Intro to Modern Analysis
Section II
Summer MMX Final Exam
Name
DIES IOVIS AD III KAL AVG MMDCCLXIII
AUC
Do all problems, in any order.
Explain all answers. An unjuified response alone may not receive full credit.
No notes, texts, or calculators may be use
1. (10 pts) Suppose that cfw_xn , cfw_yn , and cfw_zn are real sequences, and that for all positive
integers n, we have xn yn zn . If both cfw_xn and cfw_zn converge and have the same limit,
L, prove that cfw_yn also converges, and its limit is L.
Sol
Gottfried W. Leibniz
1024 Differential Way
Hanover, Electorate of Hanover
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Gottfried W. Leibniz
1024 Differential Way
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Pay to the
ord