1
Kenneth A. Ross
EL
EITARY
YSIS:
THE THEORY OF
CAL
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~ Springer
Undergraduate Texts in Mathematics
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K.A. Ribet
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Solutions for Homework #2
Math 451(Section 3, Fall 2014)
8.2a) Claim: lim n2n = 0.
+1
Proof: Given > 0, let N = 1 . If n > N , then
n2
n
n
1
n
1
0 = 2
< 2 = <
=
+1
n +1
n
n
N
Given > 0, we have exhibited N so that if n > N , then  n2n 0 < . Therefore, (
Math 451 Homework 3
(due on May 27 at the beginning of the class)
Problems to be submitted:
(1) Let xn and yn be bounded sequences of nonnegative numbers.
(a) Prove
lim sup xn yn (lim sup xn )(lim sup yn ).
n
n
n
(b) Find two bounded sequences xn and yn o
Math 451 Homework 4
(due on Friday June 6 at the beginning of the class)
Problems to be submitted:
(1) Let A R and f : A R. For > 0, dene the modulus of continuity of f
(f ) = supcfw_f (x1 ) f (x2 ) : x1 , x2 A with x1 x2  < .
1
(a) For a > 0, let A
Math 451 Homework 2
(due on May 20 at the beginning of the class)
Problems to be submitted:
(1) Let tn be a bounded sequence and let cfw_xn be a sequence such that lim xn = 0. Prove that
n=1
n
lim xn tn = 0.
n
1
(n + 1)2
(a) Show that lim tn exists.
(2)
A, the direct image of E under f is f(E):= cfw_f(x) : x
image of H under f is f^(1)(H) = cfw_x A : f(x) H.
If E
E. If H
(Thm 1.1.14) Let f: A B and g: B C be functions and let H
=
B, the inverse
C. Then we have
( g f )1 H)
f 1 ( g1 (H).
(Thm 1.3.4) (
Union (A B)
Elements in A or B or both (all)
A B = cfw_x: x A or x B
Intersection (A B)
Elements in both A and B
A B = cfw_x: x A and x B
Complement of B relative to A (A\B)
Elements in A that are not in B
A\B = cfw_x: x A and x B
DeMorgans Laws (Theorem
Subsequence (Definition 3.4.1)
Let X = ( x n ) be a sequence of real numbers and let n1 < n2 < n3 < < nk
< be a
strictly increasing sequence of natural numbers. Then the sequence X = ( x n ) given
by
( x n , x n , x n , , x n , ) is called a subsequence o
Define a subsequence
If a sequence X = ( x n ) of real numbers converges to a real number x, then any subsequence
X = ( x n ) of X
k
Let X = ( x n ) be a sequence of real numbers. Then the following are equivalent:
(i) The sequence X = ( x n ) does not co
Archimedean Property (2.4.3)
If x R, there exists
Corollary 2.4.4
If S:= cfw_1/n : n
nx
N such that x
nx
N, then inf(S) = 0
Corollary 2.4.5
If t > 0, there exists
Corollary 2.4.6
If y > 0, there exists
nt
ny
N such that 0 < 1/ nt < t
N such that n y
Definitions:
Injective
Surjective
Bijective
Empty Set
n elements
Finite
Infinite
Denumerable (countably infinite)
Countable
Uncountable
 neighborhood
Theorems/Corollaries/Axioms:
WellOrdering Principle
The set N x N is _.
Suppose that S and T are sets
Midterm Exam 2
Math 451, Prof. Roman Vershynin
Fall 2011
Read the following information before starting the exam:
You may use textbooks and your own course notes, both from this course and from
any other course you have taken. You may not use any electro
Midterm Exam 1
Math 451, Prof. Roman Vershynin
Fall 2011
Name:
Read the following information before starting the exam:
No laptops or any communication devices are allowed on the exam.
Show all work, clearly and in order, if you want to get full credit.
Math 451
Homework 6
(due on Wednesday June 18 at the beginning of the class)
Problems to be submitted:
(1) Let f be an integrable function on [a, b].
(a) Show that
b
b
f (t)dt = lim
0+
a
b
f (t)dt = lim
0+
a+
f (t)dt
a
(b) Show that there exists a c [a, b
Math 451
Final exam notes
(1) Date and location of the exam: The exam will be on Tuesday June 24. The exam will start at 4 pm in
our classroom. The exam is scheduled from 46 pm. You are allowed to bring a one sided sheet with notes.
The notes would only
Partial Solutions for Homework #1
Math 451
1.4) a) Since 1=1, 1+3=4, 1+3+5=9 and 1+3+5+7=16, one might naturally conjecture that
1 + 3 + 5 + + (2n 1) = n2 for all n N.
b) Claim: 1 + 3 + 5 + + (2n 1) = n2 for all n N.
Proof: For all n N, let Pn be the math
CONTINUOUS FUNCTIONS
Math 451
Fernando Carreon
Definition of continuous functions
(with epsilonDelta)
We consider functions f whose domain dom(f ) R f : dom(f ) R.
We say that the function f is continuous at a point x0 dom(f ) if:
> 0,
> 0 such that if
SEQUENCES
MATH 451
Fernando Carreon
Definition of a sequence and examples
A sequence cfw_sn is:
n=1
1. Intuitively: An innite list of real numbers s1 ,s2 , .
2. Rigorously: A function s : N R its elements are given by the formula
sn = s(n).
Sequences can
SEQUENCES
Part II
Sequences diverging to 1 or
1
A sequence xn diverges to innity ( lim xn = 1) if for each M > 0 there
n!1
exists a number N such that M < xn for all n > N .
M
M
M
N
N
N
Similarly, a sequence yn diverges to negative innity ( lim yn = 1) if
INTRODUCTION TO
MATHEMATICAL ANAYSIS
Math 451 Spring 2014
Fernando Carreon
Some Feedback
1. What motivated you to take this course?
2. What math courses have you taken recently?
3. Do you have any experience writing mathematical
proofs? Explain.
4. Are yo
Construction of the real numbers
using Dedekind cuts
The cut defining 2
Richard Dedekind
18311916
Real numbers= Supremums of a certain kind
of subsets of Q (cuts)
Constructing the real numbers using
Dedekind cuts
Steps:
1.
Construct a special collection
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Math 451 Midterm
Friday May 30, 10:00 amnoon, in our classroom.
Sections covered in the exam: 15,712,1720. Section 6 (construction of the real numbers) will not be tested.
The midterm will be divided in two parts:
(1) Denitions and main theorems: The
Math 451
Handout on supremums, inmums and limits.
These are some properties of supremums and limit supremums of sequences. It is important that you know
these properties and how to prove them.
(1) sup S and inf S:
(a) Denition: For a nonempty bounded abov
Math 451
Topics from chapter 3: Limits of functions and continuity
We consider only functions f : dom(f ) R where dom(f ) R.
(1) Continuous functions:
(a) A function f (x) is continuous at x0 dom(f ) if:
For all sequences (xn ) dom(f ) such that lim xn =
Math 451
Topics from chapter 2 (Elementary Analysis: The theory of calculus)
(1) Denition of a sequence: A sequence is a function s : N R. Sequences are usually denoted as (sn ) ,
n=1
where sn = s(n).
(2) Convergence of sequences: A sequence sn of real nu