MATH 521, WEEK 14:
Riemann Integration
We now move to consideration of the familiar notion of integration. We
recall from our previous experiences in calculus that integration has two
seemingly distinct interpretations:
1. Area under the curve: For a suit
MATH 521 (STOVALL). HOMEWORK 7. DUE FRIDAY, 10/28.
Throughout this homework, let (X, d) be a metric space, with X 6= . Unless otherwise
stated, all subsets of Rk are endowed with the Euclidean metric.
1. Let (sn ) be a sequence in R, and assume that sn s.
MATH 521 (STOVALL). HOMEWORK 4. DUE FRIDAY, 10/7.
1. Let E be a nonempty, finite ordered set. Prove that max E and min E exist. (Hint: Use
induction.)
2. Let x, y R with x < y. Prove that there are a countably infinite number of rationals
and an uncountab
MATH 521 (STOVALL). HOMEWORK 3. DUE FRIDAY, 9/30.
1. Let Q + iQ = cfw_q + ir : q, r Q.
a. Say that q1 + ir1 < q2 + ir2 if q1 < q2 or both q1 = q2 and r1 < r2 hold. Prove that this
defines an order on Q + iQ.
b. Define (q1 + ir1 ) + (q2 + ir2 ) = (q1 + q2
Math 521: 004
Stovall
Midterm 1
October 12, 2016
Name:
Student ID:
There are 4 problems, and a total of 8 pages.
Instructions: Solve each problem in the space provided, completely and
carefully justifying each deduction. Unless otherwise stated, you can u
MATH 521 (STOVALL). HOMEWORK 1. DUE FRIDAY, 9/16.
We avoid symbols such as , , = in very formal writing (though you may use them
in homework/exams). For instance, if P , then Q, Q whenever P , and Assume P ; then
Q, are all ways of expressing the statemen
Math 521: 002
Stovall
Midterm 1
October 12, 2016
Name:
Student ID:
There are 4 problems, and a total of 8 pages.
Instructions: Solve each problem in the space provided, completely and
carefully justifying each deduction. Unless otherwise stated, you can u
MATH 521 (STOVALL). HOMEWORK 9. DUE FRIDAY, 11/11,
1. 1.
Determine whether
the following series converge or diverge. Justify your answers.
n
P
2
1
a. n=1 n (1)n 3
P
1+cos n n
b.
.
n=1
3
2. Let pn = a0 + a1 n + + ak nk , qn = b0 + b1 n + + bj nj .
Prove t
MATH 521 (STOVALL). HOMEWORK 6. DUE FRIDAY, 10/21.
Throughout this homework, let (X, d) be a metric space, with X 6= . Unless otherwise
stated, all subsets of Rk are endowed with the Euclidean metric.
1. Prove that finite unions and aribtrary intersection
MATH 521 (STOVALL). HOMEWORK 2. DUE FRIDAY, 9/23.
1. Prove Proposition 1.15 in Rudin.
2. Let F be an ordered field. Define an absolute value on F by letting
(
x,
if x 0
|x| =
x,
if x < 0.
a. Prove that | | satisfies the triangle inequality, that is,
|x +
Lecture Notes on
Real Analysis 2 Continuity and Differentiability
Following the lectures of Dr. Zhongmin Qian
Written by Jakub Zavodny
21.3.2010, v1.2
Honour Moderations in Mathematics, University of Oxford
Hilary Term 2006
Contents
0 Foreword
3
1 Functio
MATH 521 (STOVALL). HOMEWORK 5. DUE FRIDAY, 10/14.
You only need to turn in the starred problems.
Throughout this homework, let (X, d) be a metric space, with X 6= .
1. Let E X. Let G equal the collection of all open sets contained in E. Prove that
S
E =
MATH 521 (STOVALL). HOMEWORK 8. DUE FRIDAY, 11/4.
Throughout this homework, let (X, d) be a metric space, with X 6= . Unless otherwise
stated, all subsets of Rk are endowed with the Euclidean metric.
1. If (pn ) and (qn ) are Cauchy sequences in the metri
MATH 521, WEEK 15:
Sequences and Series of Functions
There are many applications where we are interested in the limit of some
sequence or series, but where the objects are not points in Rn (or some
abstract metric space X) but are functions themselves. Th
MATH 521, WEEK 13:
Derivatives, Mean Value Theorem,
LHopitals Rule
Many applications require an understanding of the rate at which one
variable changes with respect to changes in another. For instance, we might
be interested in changes in an objects posit
MATH 521, WEEKS 11 & 12:
Functions, Continuity, Uniform Continuity
1
Functions
In the course so far, we have focused our attention on the details underlying the rational and real number systems, general set structuresespecially
metric spaces and topologya
MATH 521, WEEKS 8 & 9:
Sequences, Convergence, Cauchy
Sequences, Liminf and Limsup
1
Sequences
Metric spaces give a very nice, and very general, notion of the distance
between two objects of interest.
An obvious next step is to use this notion to quantify
MATH 521, WEEK 10:
Series, Partial Sums, Convergence
1
Series
Some of the most important applications of the results for sequences in the
real numbers are with respect to analyzing series. That is to say, summations where the elements being summed are the
MATH 521, WEEKS 4 & 5:
Metric Spaces, Euclidean Spaces
1
Further Set Theory
We will pause briefly to introduce some more notions from set theory which
will factor significantly moving forward. In particular, we will be interested
in appreciating the diffe
MATH 521, WEEK 7:
Open Covers, Compact Sets
1
Insufficiency of Open and Closed Sets
Consider the question of embedding metric spaces inside of one another.
That is to say, suppose that we have a metric spaces (X, d) and a subset
Y X, and decide to conside
MATH 521, WEEK 6:
Open and Closed Sets, Closure, Connected
Sets
1
Open and Closed Sets
We should feel happy with what we have accomplished so far. We have
identitied a formal, axiomatic way to capture the idea of distance between
two points in a set. Impo
MATH 521, WEEK 2:
Rational and Real Numbers, Ordered Sets,
Countable Sets
1
Rational and Real Numbers
Recall that a number is rational if it can be written in the form a/b where
a, b Z and b 6= 0, and a number if real if it can be written in the (potentia
MATH 521, WEEK 3:
Supremum and Infimum, Fields
1
Maximum and Minimum
Consider a subset S X where X is some ordered set. For simplicity, we
may think of X as either Q or R; indeed, most of our examples will be drawn
from these well-known sets. We identify
MATH 521, WEEK 1:
Introduction, Definitions & Review
1
Introduction
The purpose of this course is to give a rigorous introduction to the subject
of mathematical analysis.
So what distinguishes mathematical analysis from the mathematics courses
you are pro
521 Analysis I Spring 2011 Final Exam
Keep justiﬁcation short and to the point. The total is 100.
= {cc E R 2 0 < x < 1} be the open unit interval in R. Give an example of
1'. (10 pts) Let (0,1) _
0 1) such that lim $7, lies outside (0, 1) but li_mxn lies
521 Analysis I Spring 2011 Exam 3
Be sure to justify your answers. The point total is 100.
1. Let f : [(1) b] a R be continuous} and for a: E [a, 13] deﬁne another Function
9(33) 2 sup{f(t) : t E [a,a:]
(a) pts) Does there exist for each 3: E [(1,1)] a p
HOMEWORK 3: SOLUTIONS
MATH 521 FALL 2013
Problem 1: First, this fails for k = 1, as R is a eld. So let k > 1 and x x Rk . If
x = 0, then there is nothing to show, as x y = 0 for all y. So let x = 0, meaning
that if x = (x1 , ., xk ) then some xi is non-z
MIDTERM 1 (SAMPLE)
MATH 521 FALL 2013, BRIAN COOK
Remarks: This is an in class exam, with an allotted time of 50 minutes. No books, calculators, or any other items other than a pen/pencil and paper are to be used during the exam.
Do not cheat, as cheaters